Helen De Cruz' News APPS Blog post speculates on an issue that pet owners have speculated about for some time. I'm agnostic about Shakespeare's capacities, necessarily so--I do have obvious risks for bias--but it's fun to speculate. (I suspect that Shakespeare can distinguish between words based on what I say as opposed to my tone, for whatever it's worth.)
As I am reading, our cat, who happens to be called Leibnizcomes 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 classical view is that animals learn their names through classical conditioning, viz., what they learn is to *respond* to the name, not recognize themselves as such. Positive reinforcements such as cuddles and treats teaches the animal come to the owner whenever they hear their name. Similarly, the animal learns that if it is in a situation that the owner did not like in the past (e.g., trying to steal food left on the kitchen stove or opening the dustbin), it runs away as soon as it hears its name since the past conjoining of name + bad situation was negatively enforced.
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.