The number of ordered pairs of commuting functions is [A181162][1].  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 $n=7$.

  [1]: http://oeis.org/A181162

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 **at most** $n^{C(f)}$ 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 **at most** 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?  [NOTE: I added "at most" since some assignments don't work: the image of a point in a cycle of length $k$ must lie in a cycle whose length is a divisor of $k$.]