The number of ordered pairs of commuting functions is A181162. I agree with those counts up to n=6. 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 $n=6$.