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Brendan McKay
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The number of ordered pairs of commuting functions is A181162. I agree with those counts up to n=7. There is little in OEIS that helps to answer the asymptotics question.

Incidentally, the probability that $f(g(1))=g(f(1))$ is not $1/n$. I think it is $1/n + (n-1)/n^3~$ though I might have miscalculated. That formula works up to $n=7$.

Brendan McKay
  • 37.7k
  • 3
  • 80
  • 147