The analogies literally don't matter though. If it behaves "matter-like" enough, then it's matter. We generally regard this to mean entities which can have an i

The analogies literally don't matter though. If it behaves "matter-like" enough, then it's matter. We generally regard this to mean entities which can have an independent momentum and position. Hence: dark matter - because the effect we observe seems to behave that way. MOND is distinct because it declares what we're seeing is not a bunch of entities with that character. It's worth noting we also talk about a lot of things like this even though strictly speaking they're not: i.e. electron-holes in semiconductors can be perfectly well modeled as their own form of matter - they can't just leave the semiconductor environment. But they do have momentum and position. Dark matter could literally be the same type of effect, but if you're a wrinkle in space-time then it doesn't matter unless that description has some sort of experimental reality, and we currently do not observe it to do so. It is not like physicists would not rush to test any experimental formulation of dark matter as such that was accessible with current or near-future instruments.