No.

(Moderator's note: I am adding the explanation of the above answer, as provided in comments given by the poster whose account was deleted long ago, to make this a proper answer.)

Philosophically, it's a little hard to prove that something is not a coincidence, so you should take my answer to mean that I understand both and that there is no relation. Perhaps one way to think of it is as follows. The congruences proved by Ramanujan for $5$, $7$, and $11$ are very similar to each other, more so than the exceptional isomorphisms $A_5 \simeq L_2(5)$ and $L_3(2) \simeq L_2(7)$.

More generally, there is a unified reason why congruences for the partition function exist, they arise by projection onto some finite dimensional space of ordinary $p$-adic modular forms of weight $−1/2$. This space has dimension which grows roughly linear in $p$. For $p < 12$ this space is zero dimensional, hence Ramanujan's congruences. For $p$ in some further range, this space is one dimensional, explaining the congruences of Atkin. For larger $p$, this space is still finite dimensional but consists of eigenforms for the Hecke operators with different eigenvalues, so the implied congruences are not so transparent. (Nick Ramsey is writing an appendix to Folsom-Kent-Ono which will explain some of this.)

On the other hand, the exceptional isomorphisms are, well, exceptional. They differ in flavor from each other, and there is nothing that persists of this nature for larger $p$. As for what happens to $p=2$ and $p=3$, they are absent from the "exceptional" list because the groups are solvable, whereas in the case of partitions they are absent because the generating function for the partitions is $q^{1/24} \eta^{−1}$, where $\eta = q^{1/24}\prod_{n=1}^\infty (1−q^n)$ is modular form. That means one is really studying the function given by

$$1/\eta(24\tau) = \sum a_m q^m,$$

where $a_m = 0$ unless $m = −1+24n$, in which case $a_m$ is the number of partitions $p(n)$ of $n$. The methods above for $p=2,3$ give congruences for the coefficients of this series when $m$ is divisible by $2$ or $3$. Since $a_m=0$ in these cases, one does not deduce anything about congruences for the partition function $p(n)$. Putting this together, it seems that the connection between the fact that Ramanujan's congruences hold for the same set of primes as those for which $L_2(p)$ is simple is the law of small numbers.