To the answer above: "I am the being in the mirror", one has to ask themselves: Is it so strange that a cat seeing it's own paw in front of it and recognizing it as it's his OWN, and not chasing it's own tail because it knows it's his and he's not a kitten anymore?
So how much different is that when they come to understand, especially after you hold the cat several times in front of the mirror with you, making the association and he's no longer weirded out? Many cats making this transition look in curiosity and awe and move on, never really caring that much about it -- much like kids.
Bring a cat into a bathroom with a mirrored wall behind the sinks, when he's dealing with something where the mirror is in his line of sight. Walk in behind. When the cat turns around to look at you to play instantly, instead of trying to play with the mirror -- it recognizes that it shows whats behind him. I have observed this.
If a cat recognizing it's own tail out of it's "perif" is nothing special, neither is recognizing that it's him in the mirror. He may not be cognitively advanced to think deeper than "Yep, that's me, I have THIS look. Oh, am I getting fat?" -- they care less about that.
A good test would be to have a cat with whom you can put something on his head and he doesn't care -- unless it's something he HATES by its LOOK. Once you've established that over and over 100%, and have been able to put it on him without him caring without him SEEING it -- put him in front of the mirror and see if he tries to take it off.
A red dot isn't going to do squat. They don't give a shoot about things like that. If you want to know whether they know IT'S THEM, put them in position where there's something they DON'T WANT on them -- and they try to get it off... where it's something they don't mind on them, they don't try to get it off after seeing themselves in the mirror.
Not fool-proof, though. They may not mind it on them because it's purely how it looks. They can just walk away from the mirror and they don't have to worry about it.
But back to the original point: I don't think it's a big deal that they can recognize that it's themselves. To UTILIZE that for deeper thinking in making connections is another matter.