Why are those very similar languages in a different complexity class?
It's because the running time to test $k$ DFAs, each of size $n$, is something like $\Theta(n^k)$.  This is polynomial when $k$ is fixed (like 2), but exponential when $k$ is not fixed (e.g., when $k=n$, you get something like $n^n$).

(We don't actually know for sure what the best running time for that is, but we suspect it's something like that; and that will explain why those similar-sounding problems have different complexity classes.  Likewise, we don't actually know for sure that P is a different complexity class from NP, but we suspect it is.)