The Schrödinger equation emerges from classical mechanics most closely (well ok that's a bit subjective) from the Hamilton Jacobi frame work, and it was indeed here that Schrödinger saw, in hindsight, because in the beginning he pretty much guessed it, the biggest connection to classical dynamics. This is also related to the optic-mechanical relation that abstracts mechanics to the point it becomes comparable to optics.
Hamilton Jacobi theory: https://en.m.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equa...
Optic-mechanical analogy: https://en.m.wikipedia.org/wiki/Hamilton%27s_optico-mechanic...
Schrödinger equation from HJ theory: https://www.reed.edu/physics/faculty/wheeler/documents/Quant...