My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition:

The Category $\mathcal{O}$ has objects $\mathfrak{g}(A)$-modules $V$ which are $\mathfrak{h}$-diagonalizable with finite dimensional weight spaces such that there exists a finite number of elements $\lambda_1, \dots, \lambda_n \in \mathfrak{h}^*$ such that $P(V) \subset \bigcup_{i=1}^n \mathfrak{h}_{\leq \lambda_i}$, where $P(V)$ is the set of weights for $V$.

In particular, this definition allows that for any modules $V, W \in \mathcal{O}$, we have $V \otimes W \in \mathcal{O}$.

To contrast, most authors writing about Category $\mathcal{O}$ in the finite semisimple case (for example, in Humphrey's book) take as axioms

(1) $V$ is a finitely-generated $U(\mathfrak{g})$-module

(2) $V$ is $\mathfrak{h}$-semisimple

(3) $V$ is locally $\mathfrak{n}$-finite

With this choice of axioms, we no longer have that $\mathcal{O}$ is closed under tensor products.

What is the motivation/necessity for taking the differences in axioms between these two cases? Why should one not always take the formulation in Kac for "finite-type" (i.e., finite semisimple) Kac-Moody algebras?