intuition - What is a symplectic form intuitively?

Incidentally, I more or less disagree that symplectic geometry captures what I would consider "classical mechanics". The reason is that in all the examples that I think deserve to be called "classical mechanics", I actually have a configuration space $N$, and your symplectic manifold is $X = {rm T}^*N$ the cotangent bundle. Then, of course, the symplectic form is precisely (part of) the cotangent structure.

This is not to say that symplectic geometry isn't interesting — it's led to great mathematics, and certainly captures some of "classical mechanics". From the physics perspective, what I think makes it most interesting is that it shows that there are strange symmetries between mechanical systems, when you have a symplectomorphism ${rm T}^*N to {rm T}^*N'$ that does not arise from a diffeomorphism $N to N'$.

But physics is not invariant under all symplectomorphisms. Otherwise, how would I know which coordinates are "position" and which are "momentum"? And I do believe that I know this, although maybe I'm wrong. You and I should get together and compare if our Darboux coordinates differ only by a map $N to N'$, or by some more interesting symplectomorphism.