The answer is probably far from what you expect (other end of the spectrum, so to speak).
As noted in a comment, for numerical linear algebra Mathematica, at some level, uses library BLAS. I believe this does not use asymptotically fast matrix products for two reasons. One is that those methods are not able, as best I recall, to take advantage of data locality in the way that highly optimized level 3 BLAS does using traditional multiplication of matrices. The other is that the numerical stability of various algorithms is worsened by fast multiplication methods. (I think this may also be the case for ffts vs dfts, despite the former involving far fewer operations; has to do with reuse of correlated error I think).
Important caveat: Either or both of these reasons may have become invalid since last I had read anything on this topic. And regardless of what I wrote, level 3 BLAS might or might not be making use of fast multiplication.
There is at least one place where Strassen (7-for-8) multiplication is in fact used. It is when using 2x2 matrices that are needed for asymptotically fast integer gcd and continued fractions implementation code. Strassen in this case does not itself make the algorithm asymptotically faster, but since the bulk of cost is in large integer multiplications one gets a 1/8 speed boost on those parts of the algorithms that use 2x2 matrix products. If memory serves me correctly this gives an overall gain factor somewhere close to 1/8 (maybe 10% or so).