Could you accept this as an answer, so that the question is marked as resolved?

It's not clear which of two notions you're referring to. Do you mean you want the hom-categories (between two objects) of the 2-category to be categories carrying some 2-algebraic structure, or do you want the local hom-sets (between two 1-cells) of the 2-category to be sets carrying some algebraic structure? The former is what most people would call "enriched 2-categories". The latter might be called "locally-enriched 2-categories".

@MaximeRamzi: yes, I'm really interested in the question from a conceptual perspective, so presentability assumptions are stronger than I'm hoping for.

@MaximeRamzi: the case where $\mathbf C$ is presentable is already spelled out in my question.