My husband and I are making revision on a paper that mentions, at some point, briefly, Leibniz. We read the paper aloud to see if it sounds alright. As I am reading, our cat, who happens to be called Leibniz (photo), comes expectantly into the house from the garden. But what, if anything, does he understand by his own name? He also recognizes other designators, such as "cat" (in English and Dutch).
The problem with this view is that recent work has shown that animal (in particular dog and parrot - not much work on cats because they are notably uncooperative in experimental settings) language learning skills are far more sophisticated. Dogs, for instance, can fast map new words for unfamiliar objects. They do this by reasoning by exclusion: if asked "fetch the dinosaur" and presented with a heap of objects, one of which does not correspond to a word the animal knows, the dog will take the dinosaur and remember this word for months to come.
Also, animals have sophisticated conceptual understanding - more sophisticated than classical behaviorism + conditioning has it. Moreover, animals such as chimps, dolphins and even sea lions have shown capacities to learn to map symbols to concepts. Bottlenose dolphins in the wild have signature whistles to denote each other.
So if a dog can learn the word "ball" by fast mapping a linguistic expression to a concept, why would the dog not similarly learn to fast map his name to himself? Typically (this is anecdotical), dogs learn their names really quickly, and at any rate our cat learned his name within a few days. We tried operant conditioning to teach him other simple things and that took months and months.
However, the problem with this richer interpretation is that animals do not seem to have a concept of self, with perhaps the exception of corvids, elephants, great apes and dolphins and whales. They do not recognize themselves in the mirror, which is seen as a standard test for self-awareness. Dogs do not succeed in the mirror tests, neither do cats (dogs might respond to cues about themselves in other modalities like urine smell, according to Bekoff). So if that's true, an interesting asymmetry arises: a dog who helps people with disabilities, for instance, learns lots of concepts by mapping symbols onto concepts, but it learns its own name via a very different process, namely classical operant conditioning. Still, there is perhaps a way that animals can learn their own names without having a full self-awareness.
Primatologist Celia Heyes for instance argued that many animals have a body concept of self: it makes sense that you know as an agent where your own body ends and the world begins, so that you can interact, gauge whether or not it's wise to make a jump, etc. Perhaps animals learn to map their name onto this body concept of self.