I’m referring to the fact that non-human animals are widely known to have language, and that the only way the quote could make sense is some specialized meaning of language that wasn’t specified.
"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.