If so, doesn't this characterize language as being within a language game, thus undermining Wittgenstein's anti-essentialism?
There is a strong case to be made that Wittgenstein did not hold any 'anti-essentialism', for in PI he clearly rejects metaphysics (eg see here).
In any case, it would be quite wrong to think that games or gameness determines the 'essence' of language. Read on in PI and you will see a game is for Wittgenstein a 'family resemblance' concept. There is no essence of a game, but many overlapping features with no sharp edges (see http://en.wikipedia.org/wiki/Family_resemblance).
How should we explain to someone what a game is? I imagine that we should describe games to him, and we might add: "This and similar things are called 'games' ". And do we know any more about it ourselves? Is it only other people whom we cannot tell exactly what a game is?—But this is not ignorance. We do not know the boundaries because none have been drawn. To repeat, we can draw a boundary— for a special purpose. Does it take that to make the concept usable? Not at all!
In Tratactus, Wittgenstein says that no proposition can say anything about itself:
3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").
You should read this question to clarify the ideia: