Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a category internal to $\mathcal E$. If $\mathcal W$ is a locally cocomplete bicategory, then a monad in $\mathcal W\text{-}\mathbf{Mat}$ is a category enriched in $\mathcal W$. §2 of Cruttwell–Shulman's A unified framework for generalized multicategories contains a nice introduction to these ideas.
It is known that $\mathcal E \to \mathbf{Span}(\mathcal E)$ has a universal property, given by adjoining right adjoints to morphisms in $\mathcal E$ subject to a Beck–Chevalley condition.
Assume for now that $\mathcal W$ is small. There is an identity-on-objects pseudofunctor $\mathbf{Set}/\mathbf{ob}(\mathcal W) \to \mathcal W\text{-}\mathbf{Mat}$ such that every morphism in the domain has a right adjoint in the codomain (see §1 of Betti–Carboni–Street–Walter's Variation through enrichment, for instance). It therefore seems reasonable to expect that $\mathcal W\text{-}\mathbf{Mat}$ has a universal property analogous to that of $\mathbf{Span}(\mathcal E)$, but for which the Beck–Chevalley condition is replaced by a some other suitable condition.
Does $\mathcal W\text{-}\mathbf{Mat}$ have such a universal property?
In Remark 15.21 of Garner–Shulman's Enriched categories as a free cocompletion, they give a universal property of $\mathcal W\text{-}\mathbf{Mat}$ as a cocompletion of $\mathcal W$ under small $\mathbf{Coprod}$-enriched coproducts, which may be a starting point.
