Can it really be used to mean "homonym" outside the scope of the game?
People who know the game will use it informally, not in written speech and it can easily happen that people don't know the game. Personally, I will only use it among my family who played the game with me when I was a child.
Maybe it is comparable to people using "Simon says" in informal speech.
I've not known the game, until now and I've never heard anyone say "Teekesselchen" as synonym for a homonym. So I would say: no.
otherwise: maybe you can use it, because someone having a clue of homonyms will know the game.
can you try it?