In fact, no such claim is made. Chomsky's point is that, given how fast human beings acquire language, we must have some special structure in the brain that gives us this capacity, that we are born with. That doesn't exclude the possibility that other animals also possess some form of this capacity, it just excludes the possibility that Skinner's behaviorist model was right - that a being without such an 'organ' could learn language, or at least that it could do it with the speed with which humans do it.
However, Chomsky is rightly skeptical of animal language claims. One of his main discoveries from studying Universal Grammar is that (human) language is mostly a mechanism for thought, not communication. This is mostly argued from the way many concepts are encoded into language in a way that is very efficient to the one doing the encoding, but very inefficient for decoding by someone who lacks the proper context - better suited to the thinking process of the person who produced the phrase, less suited to communicating the concept to someone else.
Because of the above, vocabulary is hardly worth studying as part of language, especially when discussing concrete terms. If you truly want to claim an animal possesses something similar to human language, you need to study the way it creates phrases, and the way it can compose simple phrases into complex phrases - that is where the key power of human language lies. An animal that can recognize a million words and even basic syntax ("do A then B" != "do B then A", which is already more than most animals have been successfully taught) but that can't form novel phrases from simple phrases should not be labeled to possess the same kind of capacity that humans do.
Note that none of this is to say that possessing the capacity for language should be required to consider someone conscious or worth protecting! I absolutely believe that apes and a vast swathe of mammals (and even animals in general) posses enough consciousness that we should feel morally obligated to not torture them (I do not believe that humanely sacrificing an animal and consuming it's flesh is torture; raising a chicken in a coop where it can't move more than a few cm for its entire life is, though;;; drug testing on rats is a much more difficult moral matter).