Is there a source collecting data on the numbers of Mathematics professors (I mean with rank of Ordinarius, or ordentlicher Professor) in German Universities around 1890?
To see how clearly both the Ministry of Education and the faculty members distinguished mathematics from other subjets, and indeed different fields of mathematics from one another, see Renate Tobies Zur Berufungspolitik Felix Kleins. - Grundsätzliche Ansichten, in: Natur Technik, Medizin, 24 (1987) 2, S. 43-52.
I recommend the authoritative volume "Das Studium der Mathematik an den deutschen Universitäten seit Anfang des 19. Jahrhunderts" by Wilhelm Lorey, published in 1916. This digitalized edition can be accessed from inside the US.
As for the number of professors: to the best of my knowledge, German universities had two ordinary and one extraordinary professor, except Berlin and later Göttingen, which had three ordinary professors.
A relevant source to find answers to this questions is the book (out of print in Germany):
[Ferber 1956] Christian von Ferber: Die Entwicklung des Lehrkörpers der deutschen Universitäten und Hochschulen 1864-1954. Band 3 von Untersuchungen zur Lage der deutschen Hochschullehrer. Vandenhoeck & Ruprecht, 1956, 244 pages
which is the declared source of the data displayed in the following table:
I did not check these data. And yet, I decided to trust these number. Strictly speaking the statements implicit in the above table are not quite defined, as the time-resolution is too coarse. As a programming exercise, I constructed the following histograms, and computed three usual statistics (mean, standard deviation, coefficient of variation), pathetic as that may be given such a small sample size:
I recognize that the OP is about mathematics only, and yet, while I was at it, I fed the numbers in [Mehrtens 1990, Tabelle 3] about the subjects physics and chemistry into my program as well, obtaining the following charts, which I reproduce here, too, for comparison:
Physics:
Chemistry:
A specialized reference. The following is restricted to one German university only. In the 19th century, there was but one university in Berlin; there was also the Gewerbeinstitut/Gewerbeakademie (which roughly means 'industry/trade institute'), which had a more limited offering of courses, and was Weierstraß's first academic employer).
A standard source-book for the early history of University of Berlin is:
[Biermann 1988] Kurt-Reinhard Biermann: Die Mathematik und ihre Dozenten an der Berliner Universität 1810-1933. Akademie-Verlag Berlin 1988.
The book is in German, and is out of print, even in Germany.
This book meticulously describes the history of University of Berlin in the interval given in the book's title. In particular, the book has voluminous 'backmatter' with reproductions of original documents (such as proposals to the ministry, or reviews of theses; these are not given in facsimile though), and tables on professors and doctoral students; regarding your question, let me simply describe the 'line of succession' in its entirety, taking the liberty to interpret your 'around 1890' as the interval [1810,1933]. The following is my literal translation from [Biermann 1988, p. 343-348]; I couldn't finish the translation yet. [Biermann 1988, p. 341] says that the source for the following list is
Johannes Asen: Gesamtverzeichnis des Lehrkörpers der Universität Berlin. I. 1810-1945. Leipzig 1955
and adds that the data have been 'checked, expanded by numerous additional data, and corrected in some cases'. Biermann says that the basis for these corrections were files in an archive of the 'Philosophische Fakultät' at Berlin, some personnel files, and some university calendars. [Biermann, p. 341] says that the entries are given in 'chronological order, according to the date of the start of the first employment at University Berlin, with ties broken by rank, and ties in both date and rank being broken by alphabetical order of last name'.
$\quad$
1 Tralles, Johann Georg. Born 15 October 1763 Hamburg. Died 19 November 1822 London. Ordinarius (for mathematics and physics) since 30 May 1810. Lectured from Summer 1811 till Summer 1822. Position terminated: by death. Successor: Dirksen.
$\quad$
2 Fischer, Ernst Gottfried. Born 17 July 1754 Hoheneiche bei Saalfeld (Thüringen). Died 27 January 1831 Berlin. Extraordinarius (for mathematics and physics) since 30 May 1810. Lectured: no mathematical lectures. Position terminated: by death.
$\quad$
3 Eytelwein, Johann Albert. Born 21 December 1764 Frankfurt am Main. Died 18 August 1849 Berlin. Extraordinarius (for mechanics and mathematics) since 1 October 1810. Lectured from Winter 1811/12 till Winter 1814/15. Position terminated: by changing to the position of 'Oberlandesbaudirektor' in Berlin on 4 January 1816.
$\quad$
[...]
$\quad$
25 Weierstraß, Karl. Born 31 October 1815 Ostenfelde bei Warendorf (Westfalen). Died 19 February 1897 Berlin. Extraordinarius (for mathematics) since 11 October 1856, before and simultaneously until 1864 Professor at the 'Gewerbeinstitut Berlin'. Ordinarius (for mathematics) since 2 July 1864. Position terminated: by death. Successor: Schwarz.
$\quad$
26 Clebsch, Alfred. Born 19 January 1833 Königsberg/Prussia. Died 7 November 1872 Göttingen. Privatdozent (for mathematics) since 12 December 1872. Lectured in Summer 1858. Position terminated: by becoming professor of theoretical mechanics ato TH Karlsruhe on 22 September 1858
$\quad$
27 Christoffel, Elwin Bruno. Born 10 November 1829 Monschau/Eifel. Died 15 March 1900 Strasbourg. Privatdozent (for mathematics) at Berlin since 2 August 1859. Lectured at Berlin from Winter 1859/60 till Summer 1862. Position terminated: by becoming professor of mathematics at ETH Zurich on 5 September 1862. [not in Biermann 1988: Christoffel became Ordinarius for mathematics at the (then German) Strasbourg University in 1872]
$\quad$
28 Kronecker, Leopold. Born 7 December 1823 Liegnitz/Legnica (Poland). Died 29 December 1891 Berlin. Lecturing member of the Berlin Academy since 1 October 1861. Predicate 'Professor' since 18 March 1864. Ordinarius (for mathematics) since 11 March 1883. Lectured from Winter 1861/62 to Winter 1891/92. Position ended: by death. Successor: Frobenius.
$\quad$
29 Fuchs, Lazarus. Born 5 May 1833 Mosina/Moschin (Poland). Died 26 April 1902 Berlin. Privatdozent (for mathematics) 27 October 1865. Extraordinarius (for mathematics) 7 December 1866, with predecessor being Arndt. Lectured at Berlin from Summer 1866 to Winter 1868/69. Position ended by becoming Ordinarius (for mathematics) at University of Greifswald on 31 March 1869. Successor: Thomé. Ordinarius (for mathematics) at Berlin since 7 April 1884; before that was Ordinarius at University Heidelberg, with predecessor being Kummer. Lectured from Winter 1884/85 to Winter 1901/02. Position ended: by death. Successor: Schottky.
$\quad$
30 Thomé, Wilhelm. Born 13 March 1841 Oberdollendorf bei Königswinter/Rhein. Died 1 October 1910. Privatdozent at Berlin (for mathematics) 17 April 1869. Extraordinarius at Berlin (for mathematics) 16 April 1870, with predecessor being Fuchs. Lectured from Winter 1869/70 till Winter 1873/74. Position ended: 23 April 1874, by becoming Ordinarius at Greifswald. Successor: Bruns
$\quad$
31 Pochhammer, Leo August. Born 25 August 1841 Stendal/Saxony-Anhalt. Died 24 March 1920 Kiel. Privatdozent (for mathematics) at Berlin since 25 January 1872. Lectured at Berlin from Summer 1872 to Winter 1873/74. Position ended 21 March 1874, by becoming Extraordinarius at University Kiel. Not in [Biermann 1988]: Pochhammer became Ordinarius at Kiel in 1877, and seems to not have retired until well into his 78th year. In particular, Pochhammer was Ordinarius in the requested year 1890.
$\quad$
32 Frobenius, Ferdinand Georg. Born 26 October 1849. Died 3 August 1917 Berlin-Charlottenburg. Extraordinarius (for mathematics) 27 March 1874. Lectured at Berlin from Winter 1874/75 till Summer 1875. Position ended 30 September 1875, by becoming professur at ETH Zurich, succeeded by Wangerin. Ordinarius (for mathematics) at Berlin 16 March 1892, up until then professor at ETH Zurich; predecessor: Kronecker. Lectured at Berlin from Winter 1892/93 till Summer 1916. Position ended 3 July 1916, by retirement. Successor: Carathéodory.
$\quad$
33 Bruns, Ernst Heinrich. Born 4 September 1848 Berlin. Died 23 September 1919 Leipzig. Extraordinarius (for mathematics) 24 January 1876, before that Privatdozent at University Dorpat/Tartu (Estonia), preceded by Thomé. Lectured from Summer 1876 to Summer 1881. Position ended 31 March 1882 , by becoming Ordinarius at University of Leipzig. Successor: Hettner.
$\quad$
34 Wangerin, Albert. Born 18 November 1844 Greifenberg(Pomerania). Died 25 October 1933 Halle (Saale). Extraordinarius (for mathematics) 2 March 1876, as successor of Frobenius. Lectured at Berlin from Summer 1876 till Summer 1882. Position ended 30 September 1882, by becoming Ordinarius (for mathematics) at Halle, Successor: Netto.
$\quad$
35 (not applicable: professor of astronomy)
$\quad$
36 Hettner, Georg. Born 21 August 1854 Jena. Died 24 May 1914 Berlin. Extraordinarius (for mathematics) 11 January 1882, before: lecturer at Göttinge. Predecessor: Bruns. Ordinarius at TH Berlin snce 1 October 1894. Lectured at Berlin from Summer 1882 to Winter 1913/14. Position ended: by death.
$\quad$
37 Netto, Eugen. Born 30 June 1846 Halle/Saale. Died 13 May 1919 Gießen. Extraordinarius (for mathematics) at Berlin 6 September 1882, before: Extraordinariu at Strasbourg; predecessor Wangerin. Lectured at Berlin from Winter 1882/83 to Winter 1887/88. Position ended 31 May 1888 by becoming Ordinarius at Gießen; Successor: Knoblauch
$\quad$
38 Knoblauch, Johannes. Born 27 August 1855 Halle/Saale. Died 22 July 1915 Berlin. Lecturer (for mathematics) at Berlin 15 March 1883. Extraordinarius at Berlin 4 April 1889 (unsalaried); predecessor: Netto. Extraordinarius at Berlin 28 May 1890 (salaried); lectured at Berlin from Summer 1883 to Summer 1915. Position ended: by death. Successor: [Issai] Schur.
$\quad$
39 Runge, Carl. Born 30 August 1856 Bremen. Died 3 January 1927 Göttingen. Lecturer (for mathematics) at Berlin 9 June 1883. Lectured at Berlin from Winter 1883/84 to Summer 1886. Position ended 30 September 1886 by becoming professor at TH Hannover
$\quad$
40 Hensel, Kurt. Born 29 December 1861 Königsberg/Prussia. Died 1 June 1941 Marburg. Lecturer (for mathematics) at Berlin 2 August 1886. Extraordinarius at Berlin 7 March 1892. Lectured from Winter 1886/87 to Summer 1902. Position ended 31 October 1902 (by becoming Ordinarius at Marburg)
$\quad$
41 Kötter, Ernst. Born 7 August 1859 Berlin. Died 26 January 1922 Aachen. Lecturer (for mathematics) at Berlin 15 March 1887. Titular professor at Berlin 6 June 1894. Lectured at Berlin from Summer 1887 to Winter 1896/97. Position ended 31 March 1897 (by becoming professor at TH Aachen)
$\quad$
42 Schlesinger, Ludwig. Born 1 November 1864 Trnava (Tyrna). Died 16 December 1933 Gießen. Lecturer (for mathematics) at Berlin 7 June 1889. Titular professor at Berlin 6 June 1894. Lectured at Berlin from Winter 1889/90 till Winter 1896/97. Position ended 15 April 1897 (by becoming Extraordinarius at Bonn)
$\quad$
43 Günther, Paul. Born 2 April 1867 Bernburg/Saale. Died 27 September 1891 Berlin. Lecturer (for mathematics) 6 August 1890. Lecture announced for Summer 1891, lecture not given for health reasons. Position ended: by death.
$\quad$
44 Schwarz, Hermann Amandus. Born 25 January 1843 Sobieszów (Hermsdorf unterm Kynast). Died 30 November 1921 Berlin. Ordinarius (for mathematics) 1 April 1892, before: Ordinarius at Göttingen; predecessor: Weierstraß. Lectured at Berlin from Summer 1892 to Summer 1918. Retired 31 March 1917; Successor Schmidt
$\quad$
45 Landau, Edmund. Born 14 February 1877 Berlin. Died 19 February 1938 Berlin. Lecturer (for mathematics) at Berlin 27 June 1901. Titular professor at Berlin 29 September 1905. Lectured at Berlin from Winter 1901/02 till Winter 1908/09. Position ended 22 February 1909 (by becoming full professor at Göttingen)
$\quad$
46 Schottky, Friedrich. Born 24 July 1851 Breslau. Die 12 August 1935 Berlin. Ordinarius (for mathematics) at Berlin 1 October 1902, before: Ordinarius at Marburg; predecessor Fuchs. Lectured at Berlin from Winter 1902/03 till Summer 1926. Retired 31 March 1921. Successor Schur.
$\quad$
47 Schur, Issai. Born 10 January 1875 Mogilew/Dnepr (Mogiljow,Mahiljou). Died 10 January 1941 Tel Aviv. Lecturer (for mathematics) at Berlin 5 January 1903. Titular professor at Berlin 23 December 1909. Lecured at Berlin from Summer 1903 till Winter 1912/13. Position ended 21 April 1913 (by becoming Extraordinarius at Bonn). Extraordinarius at Berlin 1 April 1916, predecessor Knoblauch. "Persönliches Ordinariat" 29 December 1919, with salary of Extraordinarius; successor Hamburger. Ordinarius 21 May 1921; predecessor Schottky. lectured at Berlin from Summer 1916. Retired 30 September 1935
$\quad$
48 Knopp, Konrad. Born 22 July 1882 Friedenau bei Berlin. Died 30 April 1957 Annecy/Haute-Savoie/France. Lecturer (for mathematics) at Berlin 13 November 1911. Extraordinarius at Berlin 1 May 1915; predecessor Lehmann-Filhés. Lectured at Berlin Summer 1912 till Summer 1919. Position ended 30 September 1919 (by becoming Ordinarius at Königsberg/Prussia. Successor: v. Mises
$\quad$
49 Rothe, Rudolf. Born 15 October 1873 Berlin. Died 26 October 1942 Berlin. Lectureship ("Lehrauftrag") for mathematics at Berlin 1 April 1915 as Ordinarius at TH Berlin. [I don't quite understand Biermann's data here; it is unclear whether Biermann means 'lectureship as an Ordinarius', which seems oxymoronic, or whether Biermann means that Rothe had a lectureship at University of Berlin and was Ordinarius at TH Berlin, which seems more plausible] Lectured at Berlin from Summer 1915 till Summer 1919. Position ended 30 September 1919 (no reason given)
$\quad$
50 Jentzsch, Robert. Born 4 November 1890 Königsberg/Prussia. Died 21 March 1918 (fallen). Lecturer (for mathematics) 11 December 1916. Lectured at Berlin in Summer 1917. Position ended: by death
$\quad$
51 Schmidt, Erhard. Born 13 January 1876 Dorpat/Tartu (Estonia). Died 6 December 1959 Berlin. Ordinarius (for mathematics) 1 October 1917. Before this: Ordinarius at Breslau. Predecessor: Schwarz. Lectured at Berlin since Winter 1917/18. Retired 1 August 1950. Co-director of the 'I. Math. Inst.' until 22 October 1952
$\quad$
52 Carathéodory, Constantin. Born 13 September 1873 Berlin. Died 2 February 1950 Munich. Ordinarius (for mathematics) 1 October 1918, before this: Ordinarius at Göttingen. Predecessor: Frobenius. Lectured at Berlin from Winter 1918/19 till Winter 1919/20. Position ended: 31 December 1919 (by changing to Greece) [ Biermann probably means Carathéodory's change to İzmir/Smyrna to participate in the abortive attempt of founding the Ionian University of Smyrna ]
[...]
(to be continued)
Again, the above is mainly about University of Berlin only. In total, [Biermann 1988] lists 67 lecturers at University of Berlin from 1810 to 1933; of these, 18 are listed as Ordinarius.
Remarks on the definedness of the question. While the intuitive meaning is clear, strictly speaking, 'German universities' in the OP is not well-defined (more defined would be 'German speaking', or 'on the territory of the current Federal Republic of Germany', or, but that would be more restrictive, 'Prussian Universities'): in the 19th century, Germany was more united politically/administratively than in former centuries, but still much less united than e.g. in the 20th-century: in particular, Prussia and the kingdom of Saxony had a difficult relationship throughout the 19th century (and went to war against one another as late as 1866 in the 'Battle of Königgrätz' in which Saxony was allied with Austria against Prussia). The kingdom of Bavaria was yet another separate administrative entity. And yet, the Saxon university of Leipzig, and the Bavarian universities of Erlangen and Munich were 'German universities', too, though completely independently administered. To give an anecdote: a letter of Weierstraß, dated 20 December 1885, has survived in which he expresses his outrage at Klein having helped Sophus Lie to a chair at Leipzig (which, Weierstraß implies, should have gone to a German, not to a foreigner, and a young foreigner at that), and then Weierstraß (professor at Berlin) explicitly says that if Leipzig was a Prussian university, he would protest with the ministry (and implies that the Prussian ministry does not have any say in matters of Leipzig University). Such administrative independence is perhaps hard to guess given the proximity of the cities of Leipzig and Berlin. And then there are the issues around University of Strasbourg and University of Königsberg, both of which were German speaking for some time (and the latter having a role in the history of mathematics) but aren't any more.
Professors of mathematics with rank 'Ordinarius' at German-speaking universities in the requested year 1890.
To answer the OP systematically, it is necessary to first enumerate all German-speaking universities in 1890. Not all German universities which currently exist already existed back then. According to the source [Ferber 1956, p. 37], in 1890 there were the following German-speaking universities (I list the polytechnics, too, though these are not universities in the usual sense, and accordingly were called 'Hochschule' in the 19th century):
Aachen (Technische Hochschule) : (...)
Berlin: Fuchs: Ordinarius since 1884
Berlin: Kronecker: Ordinarius since 1883
Berlin: Weierstraß: Ordinarius since 1862
Berlin (Technische Hochschule): du Bois-Reymond: Etatmäßiger Professor (i.e. the analogue of 'Ordinarius' at the 'Hochschulen'; similar to the status of being a 'faculty member' mentioned in the OP) 1884 (cf. page 446 of this source)
Berlin (Technische Hochschule): Weingarten: Ordinarius since 1871
Bonn: Lipschitz: Ordinarius since 1864
Braunsberg (Lyceum Hosianum): Killing: Ordinarius since since 1882
Braunschweig (Technische Hochschule): Dedekind: Ordinarius since 1862 (Direktor 1872-75)
Breslau: Rosanes: Ordinarius since 1876
Dorpat (now Tartu; German-speaking throughout 19th century): Kneser: Ordinarius since since 1890
Dresden (Technische Hochschule): Krause: Ordinarius since 1888
Erlangen: Noether: Ordinarius since 1888
Freiburg: (...)
Gießen: Pasch: Ordinarius since 1875 (dean 1885, rector 1893)
Gießen: Netto: Ordinarius since 1888
Göttingen: Klein: Ordinarius since 1886
Göttingen: Schwarz: Ordinarius since 1875 (successor of Fuchs)
Göttingen: Schering: Ordinarius since 1868 (with regard to the OP, Schering is a borderline case, as this source explicitly says he was "ohne spezielle Zuordnung zum Fach Mathematik"; Schering is perhaps most remembered for his work as editor of Gauß's collected works)
Graz (Technische Hochschule): Mertens: Ordinarius since 1884
Greifswald: (...)
Halle: Cantor: Ordinarius since 1879
Halle: Wangerin: Ordinarius since 1882
Hannover (Technische Hochschule): Runge: Ordinarius since 1886
Heidelberg: Koenigsberger: Ordinarius since 1884 (retired 1914, resumed teaching in 1918)
Jena: Thomae: Ordinarius since 1879
Karlsruhe (Technische Hochschule): Schell: Ordinarius since 1863
Königsberg: Lindemann: Ordinarius since 1883 (changed to Münchenin 1893)
Leipzig: Neumann: Ordinarius since 1868
Leipzig: Scheibner: Ordinarius since 1868
Marburg: Weber: Ordinarius since 1884 (dean at Marburg 1887, rector at Marburg 1891; changed to Göttingen in 1892)
München: Seidel: Ordinarius since 1855
Münster: Sturm: Ordinarius since 1878
Münster: Bachmann: Ordinarius since 1868. (Bachmann is a borderline case with regard to the OP, since Bachmann left the position in the very year 1890. He was Ordinarius at Münster in 1890 though. This is an example of the 'time-resolution'-objection noted in the histograms; very strictly speaking, the case of Bachmann alone is enough reason to consider the table in [Mehrtens 1990] strictly-speaking ill-defined).
Rostock: (...)
Strasbourg: (...)
Tübingen: (...)
Wien: Weyr: Ordinarius since 1875
Wien: Stefan: Ordinarius since 1863
Wien (Hochschule für Bodenkultur): Schlesinger: Ordinarius since 1875
Würzburg: Prym: Ordinarius since 1869
