Physically animals may not be able to speak language in the exact same way we do. However, they have methods of communication that can be considered forms of language and are clearly born with that ability.
On top of that, we have taught forms of language to other primates and they've demonstrated the ability to communicate ideas and emotions, and even teach other primates. Just read up on cases like of primates being taught language like Washoe, Koko, Kanzi, etc:
Washoe (chimpanzee) - Wikipedia
Koko (gorilla) - Wikipedia
Kanzi - Wikipedia
I don't know how anyone could look at these cases and come away claiming it's impossible for animals to learn language. They clearly have the cognitive capabilities for it.