There are many different ways animals are able to communicate with each other, and with humans. Sounds, gestures, songs, body language, scents, and so on.
They can get quite creative, they can be used to communicate emotion, they can indicate that the animals have some sense for the mental state of who they are communicating with, and so on. Almost all of this is fairly obvious to anyone who's ever owned a pet (though there are plenty of cases in which there can be some subjective bias there, so it can be good to devise better tests if you're really skeptical).
But none of them have been demonstrated to have the arbitrarily complex, abstract, compositional nature that human language does; in which a finite (but large) set of words or signs can be strung together in arbitrarily complex ways, allowing a countably infinite number of different ideas to be potentially conveyed.
This is qualitatively different than any other communication method that has been demonstrated in other animals.
There's no dogma about it; it's just something that is unique to humans, as far as we know, just like the amazing ability to manipulate skin color and texture is something that's unique to cuttlefish. There are some things that certain kinds of animals can do that other kinds of animals can't, like flight, breathing via water or air, producing venoms, and so on.
It would be a pretty big scientific breakthrough to discover that there was some mammal that could change its skin like cuttlefish, or to discover that a mammal could be trained to breath water.
Likewise, discovering that non-human animal can learn a human-like language, with its arbitrary complexity through compositionality and ability to express abstract thought, is a pretty big claim too. The problem is, some people make this claim, without sufficient evidence to back it up.