Here is the definition of enrichment captured from Borceux.
My questions: It seems to me we cannot define enrichment over any monoidal category, because:
First, take the 3rd requirement, the assignment of objects of the monoidal category to each pair of objects should be wise in the way they respect the composition definition. i.e. $\mathcal{C}(A, C)$ cannot be any random object of $\mathcal{V}$, it should be the object as the result of $\mathcal{C}(A, B) \otimes \mathcal{C}(B, C)$.
Second, it needs to have enough morphisms from $I$ to every object. Because how do we know for every object there is such a map?
No one requires that $\mathcal{C}(A,C)$ be isomorphic to $\mathcal{C}(A,B) \otimes \mathcal{C}(B,C)$, just consider the fact that every category is enriched over the category of sets.
The existence of a morphism $I \to \mathcal{C}(A,A)$ for all objects $A$ is part of the data of an enriched category; no one requires that such a map exists, let alone uniquely, for every object in $\mathcal{V}$. For example, in an ordinary category one can/must specify the identity in the set $Hom(A,A)$ by the function $\{*\} \to Hom(A,A)$ given by $* \mapsto Id_A$. However, no map $\{*\} \to \emptyset$ exists.
To convince yourself that all the axioms are okay, first prove that every category is indeed enriched over $\mathbf{Set}$, and then maybe prove that the category $\mathbf{Vect}_k$ of vector spaces over a field $k$ is enriched over itself.