As far as the claims, humans keep redefining what "language" means to only include their own species. See also: intelligence. And "appearing only once" is similarly redefined as needed.
If they make their own tautologies because they're defining terms, they might be spinning beautiful poetry, but not meaningful truth.
I doubt the poetry, too, but that's a longer conversation.
"The simple understanding that one thing can be another thing is at the root of all things of our doing. From using colored pebbles for the trading of goats to art and language and on to using symbolic marks to represent pieces of the world too small to see."
I infer from this that a more sophisticated (or, in your words, specialized) definition of language that is of interest to McCarthy includes abstraction. I can talk about fire without arousing fear in a fellow human because they understand I'm not announcing the immediate presence of fire, I'm talking about it abstractly. As far as I know, no non-human animals are known to have the ability.