I think that this similarity is because both are subjected to the same type of evolution: you're exposed to the [fixed expression / meme], then as you reproduce it you're also exposing other people to the same thing. This cycle might continue indefinitely, or the thing might go extinct due to some external pressure against it; for example, competition with other [fixed expressions / memes] that seem to be more meaningful for that target audience.
(In the case of fixed expressions, there's another possible outcome: univerbation. The expression becomes a single word, and is treated as such by the speakers.)
It's also worth noting that memes are a form of communication. It might not be completely linguistic (as a good meme involves imagery), but it's still close.
[This is such a weird topic. I love it.]