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.
