Are injective structure preserving maps between structures that are known to be isomorphic always surjective as well?
In my particular case, I'm curious about injective module homomorphims between two modules that I know are isomorphic, but I'm wondering whether (if true) the statement can be generalized? I haven't been able to prove it for modules, and it seems (at least intuitively) that if a counterexample exists it would be strange. Thanks in advance!
No, there's no reason. The typical issue is an issue of size: for infinite-dimension vector spaces / modules, an injective endomorphism is not necessarily an isomorphism. For example if your base ring is $R$, take the direct product $R^\mathbb{N}$ of $\mathbb{N}$ copies of $R$, then the morphism $R^\mathbb{N} \to R^\mathbb{N}$ given by $$(x_0, x_1, \dots) \mapsto (0, x_0, x_1, \dots)$$ is clearly injective but not surjective.
You can basically reuse this example in tons of category (the map $\mathbb{N} \to \mathbb{N}$, $n \mapsto n+1$ is injective but not surjective, etc, etc). I'm not really aware of a name for a category that would satisfy the property you mention.
Even worse than that, this can fail over rings that aren't fields, even in the finitely generated case. For example the morphism of $\mathbb{Z}$-modules given by $\mathbb{Z} \to \mathbb{Z}$, $n \mapsto 2n$ is injective but not an isomorphism. Over a PID, you have a kind of converse (if $f : M \to M'$ is a surjective morphism of free modules of the same rank, than $f$ is an isomorphism), but that's not quite what you're looking for as far as I understand.