Has numerosity-preserving functions between structures been studied before?

Question

Isomorphisms are bijective functions between structures that preserve the constants, relations, and functions of the structure. I wonder, has a notion of "strong isomorphism" between structures been studied before in the mathematical literature? By strong isomorphism, I mean a numerosity-preserving function between structures that preserve the constants, relations, and functions of the structure. So, for example, the structures $(\mathbb{N};<)$ and $(\mathbb{N}^+;<)$, where $\mathbb{N}$ and $\mathbb{N}^+$ represent the set of nonnegative and positive integers respectively, are isomorphic but not strongly isomorphic, because they do not have the same numerosity, as $\mathbb{N}^+$ is a proper subset of $\mathbb{N}$. Basically, I am asking if numerosity-preserving functions have been studied.