There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.

My question is what characterizes this list?

Is there a general structure on a bicategory $B$ that we can develop a theory of $\mathcal{V}$-enriched category, for $\mathcal{V}\in ob B$ (along with base change functors) that "behave well"?

For example, if we take skew-closed categories as the enriching context, there is no yoneda lemma unless the enriching category is left normal. Therefore there should be some restrictions on the bicategory.

My motivation is that I work in one of these contexts (closed categories), and it seems the only property I really need in order to do 2-categorial constructions is the enriched yoneda lemma (the yoneda structure of $V$-categories). The details of the enriching context are not relevent to the construction, and can be removed, if we only assumed a bicategory where one can do enriched category theory.

**EDIT:**
The definition be something as follows (I'm not being precise about what notion of $3$-categories we work in as I don't know what it should be):

There is a $3$-category called $Context$ (whose objects are bicategories of enrichment contexes, with $MonCat$ being the prime example) along with a $3$-functor $(-)Cat: Context \to 3Cat$ that satisfies a bunch of axioms that make it look like the theory of enrichment along with base change.

Then there should be a sub-$3$-category of contexes whose "enrichement theory functor $(-)Cat$" gives rise to $3$-categories equipped with a yoneda structure that is compatible along base change (and other structure?).

Aside: If this sub-$3$-category exists, there can be a "completeness functor" (left adjoint to the inclusion). And in this functor for example, the completeness of $MonCat$ would probably be $SymClosedCat$ if that makes sense. For $SkewClosedCat$ the closure should be the $ClosedCat$ If we have such a completeness functor, and if the subcategory contains an initial object (which I believe exists) then we can have a very thorough description of "what is enrichment" and when two bicategories give the same information when enriching over them.

Following the comment: if we manage to define this situation, then enrichement in $n$-categories can be described using descent of $(-)Cat$ along all the forgetful functors from $nCat$ to $(n-1)Cat$

Enriched indexed categorieslays out a quite general setting for enriched cats, which might be relevant, though I haven’t time now to check through whether it covers all your examples. Martin Lundfall’s masters thesisModels of Linear Dependent Type Theorypresents of a slightly less general but still quite general setup, in terms of linear logic. $\endgroup$ – Peter LeFanu Lumsdaine Aug 28 '18 at 6:46