The matrix direct sum doesn’t look anything like a matrix direct product, yet (according to Mathworld) the group direct sum is the same as the group direct product if the groups are abelian.
So are these just two completely different usages of “direct sum”, or what?
The problem is that you’re looking at objects in one case and morphisms in the other. Category theory comes to the rescue again.
The category of abelian groups is the class of all abelian groups and the set of abelian group homomorphisms hom(A,B) from A to B.
The product (in any category) of two objects A and B is an object C with morphisms p[sub]A[/sub]:C->A and p[sub]B[/sub]:C->B such that any other object and pair of morphisms from that object to A and B has a unique map to C factoring p[sub]A[/sub] and p[sub]B[/sub]. The coproduct is defined dually: an object C with maps i[sub]A[/sub]:A->C and i[sub]B[/sub]:B->C so that any other object and pair of morphisms from A and B factors through C uniquely.
For an Abelian category (such as that of abelian groups – Z-modules – or even for R-modules over any ring R) these two objects are naturally isomorphic. In this case, the product is called the direct sum.
A matrix with entries in R is really a morphism from R[sup]m[/sup] to R[sup]n[/sup]. The direct sum of two matrices is the action of the direct sum functor in the category of R-modules. I think what you may mean by “direct product” of matrices is more properly called the “Kronecker product” or “tensor product”, which is a different operation.