Are canonical transformations (in the sense of Hamiltonian mechanics) morphisms for a certain category?
They seem to fit the archetypal description of morphisms being "structure-preserving maps".
Are canonical transformations equivalent to symplectomorphisms? (The canonical equations seem to suggest as much.) I.e. is "canonical transformation" the physicist jargon for what a mathematician would call a symplectomorphism?
If so, then does that mean that canonical transformations are just the morphisms of the symplectic category? (Called the Weinstein symplectic category in nLab.)
Do we lose any generality by only considering symplectic manifolds, or are those implicitly the only objects under study in Hamiltonian mechanics? https://ncatlab.org/nlab/show/Weinstein+symplectic+category
Yes, canonical transformations (which are defined on the phase space $T^*\Bbb R^n$) are the same as symplectomorphisms - usually a canonical transformation is a diffeomorphism of $T^*\Bbb R^n$ that pulls back the canonical 1-form $\sum p_i dq_i$ to itself plus some exact 1-form. This is entirely equivalent (because $H^1(T^*\Bbb R^n) = 0$) to saying that it pulls back the symplectic form $\sum dp_i \wedge dq_i$ to itself - that is, that it's a symplectomorphism. For reinforcement, see page 18 of McDuff-Salamon: there they say "In the classical literature on Hamiltonian mechanics, symplectomorphisms are sometimes called canonical transformations."
But you should be warned about thinking about a "symplectic category". The first naive notion of such an object is as you say, manifolds and symplectomorphisms; but this doesn't allow interesting maps between different symplectic manifolds - this category is what's called a groupoid. (All morphisms are isomorphisms.) A second attempt might be to take symplectic manifolds and morphisms $f: (M,\omega) \to (N,\omega')$ if $f^*\omega' = \omega$. But this is again very restrictive: these maps are always immersions, for instance, and therefore cannot decrease dimension, or plenty of other interesting things one might like to do.
You somehow want a way of going between legitimately different symplectic manifolds, especially for the purposes of symplectic field theory, which assigns an object (the Fukaya category) to each manifold; you'd like a way to go between these objects for different symplectic manifolds in an interesting way. You do this by Lagrangian correspondences. This is inspired by the fact that given a symplectomorphism $(M,\omega_1) \to (M',\omega_2)$, the graph is a Lagrangian submanifold of $(M \times M', \pi_1^* \omega_1 - \pi_2^*\omega_2)$. So you define a morphism to always be a Lagrangian submanifold of this product. These don't compose, so Wehrheim-Woodward generalized them to the inspirationally named "generalized Lagrangian correspondence". But these are much more general than symplectomorphisms.