Can it really be used to mean "homonym" outside the scope of the game?
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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?