This is actually meant to be a comment, not an answer, but I'm new here and I don't have enough reputation to post a comment yet...sorry!

I just wanted to note that Dickson's conjecture would imply infinitely many consecutive numbers with the same prime signature.

For example, Dickson's conjecture would say that there are infinitely many k such that 4k+1 and 9k+2 are both prime.  For each such k, 4(9k+2) and 9(4k+1) would be consecutive numbers with the prime signature (1,2).  One might expect there to be roughly $\frac{3N}{\log 4N \log 9N}$ values of k between 1 and N such that 4k+1 and 9k+2 are both prime.