As far as I can tell, many questions about topological games are currently tagged infinitary-combinatorics. (Topological games are the infinitary games I am at least a bit familiar with, although there are certainly some other infinitely long games). For example, you can find some such questions if you search for [infinitary-combinatorics] game is:q.
Since study of various infinitary games seems mostly related to set theory, to me this seems to be a reasonable tag.
I am a bit reluctant about creating an entirely new tag. Mainly because there is a limit of five tags per question - so tagging might become problematic if we have too many very specialized tags.
So my proposal would be:
Certainly there are several users on this site whose area of expertise is close to infinitary combinatorics and who know much more about this topic than I do. So certainly suggestions by experts in this area should have more weight. (And I hope some of them will join the discussion.)
But no matter what the outcome of this discussion is, we should then clarify the usage of the relevant tags in the tag info. (For example, it should be clarified whether (infinitary-combinatorics) should include questions about infinite games. Or if a new tag for (infinite-games) is created, the tag info should explicitly say whether topological games belong under this tag or not.)