That is certainly the conventional belief.

But that conventional belief is founded on a single fundamental test (the mirror-dot test) that may simply be difficult to apply to cats.

Other animals were thought not to be able to recognise themselves until they passed that test.

When they pass the test, we say that animals of that species can recognise themselves in the mirror. It's a single observation.

So the question then becomes: are we so sure about that test?

If you have not watched the video, watch it a few times.

The only other really sensible explanation is that the cat only gets this far:

The mirror cat does everything I do! Every time! I know the thing on the wall has the mirror cat in. I can only see the mirror cat's ears... Ears are interesting. Do my ears look like that? Since the mirror cat matches my every movement, that must be what my ears look like. So if I rub my paws on my ears, and watch the mirror cat copy me, I should be able to confirm that interpretation...

The cat is definitely performing a slow, precise movement in order to watch what happens in the mirror. She sits up to make sure she can see it, before she does it. (She doesn't need to sit up to clean her own ears without looking). It's far too deliberate and studied; we know what cats look like when they are concentrating on movement.

That she comes up short of realising that the mirror cat is herself is definitely plausible. But even then it is demonstrating an enhanced theory of mind, because it involves predicting that the mirror cat's actions will always match her actions, that this can be used to test an idea, and that conclusions can be drawn about what she must look like when doing the same actions.

At some point, we have to accept that the idea that we are wrongly projecting our unique intelligence onto other animals is actually pretty arrogant.

I suppose. But I think this point is in danger of being overstated. Mostly animals don't see their own reflections in water; they certainly see others' reflections in water. But when you're close enough up to see a full reflection of your face in water, what you largely see is a shadow.

Yes, the shadow has a reflection in it, but I suspect that is quickly disregarded without much contemplation of what it means.

Mirrors are another matter entirely. You can really interrogate a mirror by seeing your own reflection at considerable distance, which is usually not possible with water.

Plus you can do so at leisure, because you are not necessarily prone at that point, whereas a cat at the waterline is at risk of attack.

So I don't know if, in the scheme of things, much about reflection has had a significant impact, until the mirror.

Edit to add: the fact that some intelligent birds are able to pass the mirror dot test might make sense in this concept of how easy a reflection is to comprehend. Birds have after all had millions of years of opportunities to interrogate their reflection in water at different heights, and to see a much clearer and more complete image of their own reflection. For a bird, still water really could be a mirror.