I found a useful reference that might help answering this question (this doesn't constitute an answer to the questions in post). The reference is from p. 46-47 of the book "From Riemann to Differential Geometry and Relativity" - in those pages the author cites a 1863 letter from Enrico Betti to another mathematician:
What gave Riemann the idea of ...
There is a nice book called Princeton Companion to Mathematics (written by many authors, edited by Gowers) which explains many parts of modern mathematics, and has
a chapter on development of principal concepts.
In Gert Schubring's translation from 2016, the exact statement is:
"Every person believes that he knows what a curve is until he has learned so much mathematics that the countless possible abnormalities confuse him." (p. 199).
Your question is too broad, I think.
One nice scholarly book (which covers only the changes around 1850 to 1910, as mathematics went from a mishmash of physics and psychology, and became an abstract deductive science):
Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton University Press, 2008
I will skip the pre-history of solving polynomial equations and factoring polynomials. Let me mention that the analogy between long division of numbers and polynomials goes back to medieval Islamic mathematician al-Samawal, see Who invented short and long division?, and the Euclidean algorithm for polynomials was optimized by Hudde, a younger contemporary of ...