How did the shift from the times of Thales, Pythagoras, and even Euclid, where mathematical objects were found, exhibited, or constructed from given entities, to modern mathematics occur? In modern mathematics, entities no longer need to be constructed or computed in order to be named and manipulated; they simply need to exist. What was the transformative change that enabled this level of abstraction to be embraced and widely adopted, namely the distinction between mere existence and actual realization?
TL;DR It is important to distinguish different senses of "constructiveness", the technical one, as in constructivism, and the one of approach to mathematics as primarily problem-solving. They followed two different lines of development that are often confused (this is where Wikipedia goes wrong), and only the second underwent a shift in 19th century. The first one, on the other hand, shows gradual expansion of construction tools since Euclid and is continuous through the 19th century.
The problem-solving ("constructive" in a different sense) attitude also goes back to Euclid and dominated the field up to the end of 18th century. It gave way to the more meta-theoretical "conceptual" attitude even before Cantor, although he championed it along with Dedekind and Weierstrass, and it was canonized by Hilbert, Zermelo, etc., in the modern axiomatic method. The rise of impossibility and "pure existence" results to prominence accompanied it, but it was not directly related to non-constructivism in the technical sense.
Here is Lützen describing the shift in Why was Wantzel overlooked for a century? to explain the fate of Wantzel's now famous impossibility proofs:
"Indeed, during the 18th century, mathematics was a constructive enterprise consisting of finding solutions to problems, and this view of mathematics was still shared by most of Wantzel’s contemporaries... In the history of the quintic, Abel explicitly tried to change this view of impossibility results. As emphasized by Sørensen, Abel suggested that one ought to rephrase the problem as a problem that always has an answer.
Instead of asking for a solution by radicals, one should first ask if the quintic has a solution of this kind, and only if the answer turns out to be yes would one proceed to the question of finding such a solution constructively. By formulating the existence question as a mathematical problem, Abel had allowed the impossibility result to be a solution of a problem, not just a statement about the impossibility of finding an answer to a problem. Thus impossibility statements became proper theorems in mathematics. Several younger mathematicians continued along the same line as Abel. Evariste Galois is the most obvious example, but also Joseph Liouville’s investigations of integrability of functions and differential equations in finite form follow this lead.
These new tendencies relative to impossibility results went hand in hand with a new focus on existence theorems, and can be viewed as two sides of the same coin. In particular, Cauchy insisted that before finding the sum of an infinite series or the solution of a differential equation, one should prove their existence. And both trends were parts of a change from a constructive to a conceptual paradigm in mathematics. Another instance of this change was the emergence in the 1830s of Sturm–Liouville theory where the (qualitative) properties of solutions to certain differential equations were investigated without solving the equation in analytic form."
The above does not apply to constructive/non-constructive in the technical or even more relaxed sense, which does not necessarily track problem-solving vs theory-building. If we go by the modern definition of constructive proof, which discards proofs by contradiction = reductios ad absurdum, then the non-constructive goes all the way back to ancient Greeks. Euclid employs reductios not only in incommensurability proofs, but even in the most elementary proposition I.6, the isosceles triangle theorem, which surely goes back to Pythagoreans if not Thales.
And even if we focus just on straightedge and compass constructions, what changes over the centuries is mostly the range of construction tools. Euclid's constructionism is not that different in spirit from modern one, where axioms of set theory are used as construction postulates. Most ZFC axioms (specification, pairing, union, powerset, replacement, choice) are even formulated as such postulates. Even employment of the axiom of choice to produce sets to specifications, the very paradigm of non-constructive, is unabashedly called "construction", as in Vitali construction of Lebesgue non-measurable sets, etc. See Rodin's Doing and Showing on structural analogies between Euclidean and set-theoretic constructionism.
Already in Euclid, there are tricky cases where he implicitly relaxes his straightedge and compass strictures. The proof of proposition XII.2, which applies the so-called "method of exhaustion" to the circle quadrature, essentially presents a re-construction of the circle by transcendental means (as a limit of inscribed polygons), and then uses a reductio to prove its equivalence to the elementary compass construction. Archimedes used the "method of exhaustion" on conic sections and even solids, and others happily intersected transcendental curves (quadratrix, spiral) with lines. Unlike the modest (and omitted) completeness assumptions needed in the Elements, these "constructions" implicitly rely on strong forms of completeness for the continuum. See Bos, Construction of equations, 1637-1750 on how transcendental methods were gradually accepted in geometry.
By the 17th century, algebra and number theory also matured, so "constructions" involved infinite decimals (Stevin) and then infinite series and products (Wallis, Newton, Leibniz). Algebraic analysis, where algebra was transferred to functional infinite sums and products, ruled the 18th century. Euler's experiments with infinite "constructions" are well-known, while Fourier's "constructions" with trigonometric series were even seen as "revolutionary" at the turn of 19th century. Many more functions were thereby made "constructible", see Bressoud, Radical Approach to Real Analysis. Both lines of development paved the way to Cantor's innovations, his work on sets was motivated by Fourier series and influenced by the emerging conceptual attitude.
This change apparently occurred in the middle of 19th century.
Let me cite Carl Ludwig Siegel about it:
It is completely clear to me which conditions caused the gradual decadence of mathematics, from its high level some 100 years ago, down to the present hopeless nadir. Degeneration of mathematics begins with the ideas of Riemann, Dedekind and Cantor which progressively repressed the reliable genius of Euler, Lagrange and Gauss.
and he continues:
Through the influence of textbooks like those of Hasse, Schreier and van der Waerden, the new generation was seriously harmed, and the work of Bourbaki finally dealt the fatal blow.
C. L. Siegel, Letter to A. Weil, 1959. German original is cited in the lecture of H. Grauert, in: Mathematics and Theoretical Physics, 1993, ed.: Minaketan Behara, Rudolf Fritsch, Rubens G. Lintz, W. de Gruyter, 1995.
Similar views about the influence of Riemann on mathematics were shared by many mathematicians of that time, for example by Chebyshev and Lyapunov.
Bourbaki also acknowledges the role of Riemann in reshaping of mathematics:
Riemann has to be considered the founder of topology as well as of most of the branches of modern mathematics...
N. Bourbaki, Topology I, Historical essay.