Direct summation (taking a $p \times p$ matrix $A$ and a $q \times q$ matrix $B$ and returning a block-diagonal $(p+q) \times (p+q)$ matrix $A \oplus B := \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$) also works:

$$\det(A \oplus B) = \det(A) \det(B).$$

One can debate whether this operation deserves to be called a "matrix product", though (for instance, it is not distributive over addition).

EDIT: Another (somewhat trivial) example is the reversed multiplication operation $(A, B) \mapsto BA$. More generally, if there was a linear automorphism $T$ on $Mat_n$ that preserved the singular variety $\{ A \in Mat_n: \det A = 0 \}$, one could conjugate the usual matrix multiplication operation by $T$. In the above example, $T$ is the transpose operation $T: A \mapsto A^t$. As another example, one could let $T$ be a left multiplication operator $A \mapsto SA$ for some invertible $S$, in which case the matrix multiplication operation becomes $(A, B) \mapsto ASB$, which also seems to work. One can combine the two and obtain another operation $(A, B) \mapsto BSA$. I'm not sure if these are the only examples that can be constructed by this method.

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Direct summation (taking a $p \times p$ matrix $A$ and a $q \times q$ matrix $B$ and returning a block-diagonal $(p+q) \times (p+q)$ matrix $A \oplus B := \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$) also works:

$$\det(A \oplus B) = \det(A) \det(B).$$

One can debate whether this operation deserves to be called a "matrix product", though (for instance, it is not distributive over addition).