I've read that cats think of themselves as cats, of course. But they also think of their owners as cats.
No, instead we are just their servants. 
It's much funkier than that. They obviously can tell the difference between a human, a cat, and a dog; but they don't seem to grasp the difference in
behaviour. Research does also indicate they do understand many human words, just don't care.
A good example is BilliSpeaks on Youtube: She seems to understand pretty complex human concepts and even "soon" and "later" (and coffee = "catnip water"), but I don't think she can tell the difference between "play" and "hunt". I don't think
any cat can.
To me, it indicates cats' model of "self" and "other" is quite different to those of humans, monkeys, and some birds (see
mirror neuron). It also makes a lot of sense for it to differ, as most of the domestic cats' social behaviour, including cat colony behaviours, has evolved in interaction with humans, i.e. during "domestication".