I'm guessing it just means the same as "play opposite", in other words the character your character is threading with/playing with would be who you're playing against.
So if you play Fred Flintstone, and were looking for a storyline with a Wilma, then you might place an ad that says something like "Fred Flintstone, looking to play against a Wilma!" Or something similar, unless there's a specific usage of the term that I'm not aware of, which is possible.