I guess I've always just assumed it refers to some feature that's uniquely human—notably, recursive grammars.
And recursion as the unique trait for human language differentiation is not necessarily completely consensual https://omseeth.github.io/blog/2024/recursive_language/
Also, let's recall that in its broader meaning, the scientific consensus is that humans are animals and they evolved through the same basic mechanism as all other life forms that is evolution. So even assuming that evolution made some unique language hability emerge in humans, it's most likely that they share most language traits with other species and that there is more things to learn from them that what would be possible if it's assumed they can't have a language and thoughts.
It seems that the second link may indicate otherwise but I'm still pretty skeptical. This requires extraordinary evidence. Furthermore there may be a more practical limit of "stack size" or "context size" that effectively exceptionalizes humans (especially considering the size and proportional energy consumption of our brains).
Other animals have cognitive processes, and languages, or at least it seems to be something scientifically consensual. Thus the surprise reading the kind of statement given in introduction.
Whether humans have exceptional language habilites or even "just" a biggest board to play on with the same basic facilities seems to be a completely different matter.