Who else can recognise their own reflections?  

Well, we wouldn't know, would we?

But for the case of the monkeys, might they not have thought that it was another monkey in the mirror? They might be just practising their "monkey see, monkey do"! :P

There are other creatures who can do that, yes. I can't tell you an exact list, but I'm sure elephants are one of them - I heard about this piece of news last year, if I'm not mistaken.

And the way that the research was done was very interesting. They used a large mirror, and painted a red mark on the elephant's forehead. When the elephant say itself in the mirror, it tried to touch on its own forehead the red mark it had seen on the reflection.

(Of course, they also had a control group where the elephant was marked with invisible paint, therefore eliminating the possibility that the elephants were basing their behaviour on touch instead of sight).