The tensor product of two monoidal categories

The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.

Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.

If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $\mathcal{M}$ and $\mathcal{N}$. The Deligne tensor product $\mathcal{M}\boxtimes\mathcal{N}$ does inherit braided monoidal structure from $\mathcal{M}$ and $\mathcal{N}$ if these are braided monoidal.

If you're acquainted with $\infty$-categories you might also be interested in the tensor product of presentable $\infty$-categories: it's a real tensor product, in that it provides an equivalence $$ \begin{array}{cr} C_1\times\dots\times C_n \to D & \text{cocont.}\\\hline C_1\otimes\dots\otimes C_n \to D \end{array} $$ between the colimit-preserving functors from $C_1\times\dots\times C_n$ to $D$ and the functors from the tensor product to $D$.

There's plenty of reasons why "multilinear" gets categorified in "cocontinuous"; also, presentable categories all are cartesian (because they admit finite products, and all other shapes of limits for that matter), so they are not the most general monoidal categories, but their tensor product is fairly well understood.

Hope this helps!