Not a complete answer, but there is a surprisingly general generalization of the dimension of a finite-dimensional vector space available in any (braided?) monoidal category. In any such category, there is a notion of dimension of a dualizable object $c$ given by the trace of the identity endomorphism $\text{tr}(\text{id}_c)$. It takes values in $\text{End}(1)$ where $1$ is the monoidal unit and behaves as expected under tensor product. In the category of finite-dimensional vector spaces over a field $K$ it gives the image of the dimension in $K$.
Not a complete answer, but there is a surprisingly general generalization of the dimension of a finite-dimensional vector space available in any (braided?) monoidal category. In any such category, there is a notion of dimension of a dualizable object $c$ given by the trace of the identity endomorphism $\text{tr}(\text{id}_c)$. It takes values in $\text{End}(1)$ where $1$ is the monoidal unit and behaves as expected under tensor product. In the category of finite-dimensional vector spaces over a field $K$ it gives the image of the dimension in $K$.