Yes most other loops that produce the same result wouldn't be recognized, just this very specific, note the steps:

// step 1: Check if the loop-back branch is in desirable form.

// step 2: detect instructions corresponding to "x2 = x1 & (x1 - 1)"

// step 3: Check the recurrence of variable X

// step 4: Find the instruction which count the population: cnt2 = cnt1 + 1

It's very, very specific. But they allow that the loop has something more, see the resulting transformation at the end:

   // After this step, this loop (conceptually) would look like following:
   // newcnt = __builtin_ctpop(x);
   // t = newcnt;
   // if (x)
   //  do { cnt++; x &= x-1; t--) } while (t > 0);

This can be later potentially optimized more, in the given simplest case the rest is removed by the later optimization steps.