As I understand it, a symplectomorphism of vector spaces (resp. smooth manifolds) is required to be an isomorphism (resp. a diffeomorphism), while there is the broader category of symplectic embeddings $f : (V, \omega) \to (W, \omega')$, which satisfy $f^\star \omega' = \omega$, but need not be isomorphisms.
Is there a good reason for reserving the term symplectomorphism for only the isomorphisms? It seems to me that the category of symplectic manifolds and symplectic embeddings is richer and more natural.
This is an excellent question, and the reason behind this is largely historical: symplectomorphisms were orginally called canonical transformations and arise from classical mechanics where they are thought of as changes of variables in phase space that preserve the structure of Hamilton's equations: hence they should be diffeomorphisms.
You are right that from a modern perspective it might be more natural to use a word like symplecto-isomorphism to refer to symplectomorphisms and reserve the word symplectomorphism for the larger class of symplectic embeddings, but sadly that's not the convention.