I feel like the notion of a linear transformation between vector spaces is not the "real" notion of a morphism between vector spaces. The restriction imposed - that $V$ and $W$ be over the same field, feels like a restriction that we wouldn't want to actually make.
Instead, why not allow $F_1$, $F_2$ to be different fields, and have an accompanying homomorphism $\varphi$ between the fields? That is, define a linear map $T: V \rightarrow W$ to be a map such that $T(cx+y) = \varphi(c)T(x)+T(y)$.
I feel like this is "really" the morphism of vector spaces, and that we just ignore the subtlety. It can't be claimed that $F$ is intended as "up to isomorphism" because the standard definition of a linear map precludes that possibility.
I do have concerns about it, though. Mainly: does a bijective $T$ force a bijective $\varphi$, thus generalizing "isomorphic vector spaces" to be spaces which have the same dimension and are over isomorphic fields? That's what I'd want to happen. I can show that injectivity of $\varphi$ is forced (if $V$ is not the zero vector space) but surjectivity isn't immediately obvious to me.
My observation also extends to modules in general - I just originally thought about it in terms of vector spaces.
My attempts to answer this question myself:
1.) It's a pointless generalization - it doesn't add anything new to the study of vector spaces.
2.) There is an issue with doing this that I'm just not seeing.
3.) This generalization is done, I just haven't come across it yet.
4.) This is the "real" notion of a morphism of vector spaces, but we restrict to vector spaces over the same field largely for pedagogical reasons (linear algebra is often learned before fields/rings).
Thoughts?
Thank you