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 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.