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 haven't 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?
I wasn't aware it's a "game" (other than noun being a "game" with the goal of listing as many nouns as you can think of, and verb being a "game" with the goal of listing as many verbs as you can think of, etc.), but the term was routinely used where I went to elementary school in South-Western Germany many years ago and seemed to be familiar to elementary school teachers among my relatives from Northern Germany, as well.
I am not sure Teekesselchen is an exact synonym for homonym, though; I always felt ein Teekesselchen already denotes the pair of homonymous words.