Yes, Apple games can belong in multiple genres and it was easier to do it this way, but I'm probably going to make it a dropdown since it's more usable.
Possibly make it a tag selector for easy multiple selection. You could then leave it a text input that sort of auto completes a tag name and adds it to the search filter
That's a good idea, but then I worry that it would narrow the results too much. Not that big a deal, since you can just remove a tag again, but I'll have to play around with it and see.
A multi-select box for this would be nice. Select2 should be easy enough to implement, as you're already using jquery.
Even HTML 4.01 had <select multiple>, no jQuery required: