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 $K$.
Most notably, this notion of dimension includes as special cases several types of Euler characteristic. For example, the dimension of a bounded dualizable chain complex (I think this is equivalent to: bounded complex of finitely-generated projective modules) is its Euler characteristic, as is the dimension of a dualizable object in the symmetric monoidal category of dualizable spectra. A nice exposition is given in Ponto and Shulman's Traces in symmetric monoidal categories, which in particular describes how to use these ideas to understand the Lefschetz fixed point theorem.
