I have seen a couple of times that concepts specific to particular structures that seem "sensible" but still somewhat arbitrary can be derived from a general canonical principle at a higher level of abstraction. E.g. if I'm not mistaken, the "compatibility" of the group structure of a Lie group with its topological structure, can be derived from projecting functorially the axioms of group theory onto the category of topological spaces.
A vector space $(V,F,v_0,+_v,\cdot_v)$ can be seen as a commutative group $(V,v_0,+_v)$, together with a field $F$ and a scalar product $\cdot_v$ such that "the group $V$ is 'structurally compatible' with the the field $F$".
Is there a way to state this idea of "structurally compatible" at a higher level of abstraction, such as category theory or mathematical logic, and then derive the specific axioms of vector spaces from that?