But we are only talking about it because we can observe it. Or rather observe how it interacts with other things, but that's just semantics: all observation happens via interactions.

Dark matter could be a neighboring universe interacting with ours or something like that where the actual particle causing it is fundamentally out of reach (if it even is a particle), but just the fact that it interacts with our universe means we should in principle be able to interact with it and characterize it with experiments.

Thats more along the lines of what I was thinking - we can observe this specific interaction but without also observing other interactions it seems very difficult to define exactly what it is. Or maybe it is just that our current frame of reference (both physically and in terms of time) is just a bad vantage point to do so.

An important property of dark matter need to fit what we are seeing is that it also doesn't interact with itself. It would seem weird if it can't interact with itself because of some innate property, but we can't interact with it because we are somehow out of phase with it.