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 $n=7$.
ANOTHER relevant fact: If $f$ is a permutation, then any function $g$ commuting with $f$ is determined by the image of one element of each cycle of $f$. So the number of such $g$ is $n^{C(f)}$ where $C(f)$ is the number of cycles of $f$. Random permutations have on average only $\ln n+O(1)$ cycles, so the probability of a random function commuting with a random permutation might be something like $n^{-n+\ln n+O(1)}$ (which is an abuse of expectations but might be something akin to the truth). Is a random function more or less likely to commute with another random function or with a random permutation?