When solving a linear system, x and b only have one column. So unless the algorithm is general enough to not have that limitation, it won't be applicable to general matrix multiplication.
> When solving a linear system, x and b only have one column
Isn't it linear for whatever amount of columns?
The extension to matrix-matrix is trivial, since a matrix is just a concatenation of columns.
A matrix is a concatenation of columns yes, but how does that help us multiply matrices by using a sparse linear-system solver? Perhaps it's trivial but I don't see it.