(My understanding of how physicists use the term "momentum": they use it to describe a particular precisely defined physically meaningful quantity, which plays an important role in physics. This seems like an entirely reasonable, even admirable thing. But there seems to be a tone of disapproval in what you say. The obvious complaint about mathematicians' use of "game" would be something like: they took an ordinary word with an ordinary everyday meaning and used it to mean something weird and technical. But that's absolutely not what physicists do with "momentum"; the everyday uses are derived from the physicists' term.[1] So I'm confused.)
[1] There are actually pre-physics meanings of "momentum" but they are long dead . E.g., it could once mean 1/40 of an hour.
To save everyone else a click (though the video is short and fun): the video laments the way that "momentum" in physics refers to a wide variety of concepts: the simple p=mv in Newtonian physics, the generalized versions you get in Lagrangian and Hamiltonian mechanics, the relativistic 4-vector version, the momentum operator in QM, etc., and also (this is kinda-sorta a different sort of difference) angular as opposed to linear momentum.
I'm not sure the analogy quite works.
What's going on with the physicists' notions of momentum is that if you start with the simple p=mv that goes back to Newton, you can (1) generalize it, getting the Lagrangian and Hamiltonian notions (of which "angular momentum" is then another special case), and (2) see that Newtonian mechanics isn't actually quite how the world works, and look for corresponding notions in more accurate theories, getting the quantum and relativistic versions.
In mathematics, the main notions of "game" are what you might call the Conway type and the von Neumann type[1], and those are much more different from one another than any two notions of "momentum" in physics. Then there are other things like the idea of a "topological game" which I guess you can kinda connect to those but is effectively a third largely unrelated thing.
[1] I do not recommend taking those names too seriously.
So the physicists (in this case) are looking at a single phenomenon and deepening and broadening their understandings of it, whereas the mathematicians (in this case) are looking at a variety of really quite different things that happen to have the same name.
Anyway, if I wasn't imagining the note of disapproval in your original comment, it may be worth saying that I don't think there's anything wrong with any of this. It's perfectly normal for words to have a variety of variously-related meanings. To jam something means to stick it into a tight place. Jam is a sweet gloopy thing made from fruit and sugar. If you're "in a jam", you're having difficulties that needn't involve any physical tight place and almost certainly don't involve fruit or sugar. If someone's "jammy" they're lucky in ways that, again, probably don't literally involve tightness or fruit. If you have a "jam session" you're getting together with other musicians for improvisatory fun. All of these meanings are ultimately related to one another but they're very different things. That's just how language works. Physicists' uses of "momentum" or mathematicians' uses of "game" aren't really any different.
(The normal everyday non-mathematical use of "game" is actually a famous example of this sort of thing: there are many things we call games, and probably no single definition that covers them cleanly; rather, we learn lots of examples of things that are "games" and things that aren't, and then we call something a "game" when it resembles other things that we have learned to call games.)
And it's perfectly normal for physicists and mathematicians to look at something that appears simple and find ways to make it more and more general, deep, or precise, usually at the cost of being increasingly difficult to understand. That's just what physics and mathematics are, and so far as I can tell there's no way to get the generality and depth and precision without the difficulty.
