I may be mistaken, but I have the impression that someone is solving a similar but different question (What is the probability that two random integers are relatively prime?)

Not me.

But this question asks for (essentially) the probability of something a bit different: that gcd(i,j) = gcd(k,m), for four random integers i,j,k,m.

Indeed. But [... blank lines follow for the usual reason ...] ... one way to solve the original problem (the way I used, in fact, and it seems a pretty good way to me) goes via the answer to the simpler and more familiar one. -- g