While they can often amount to the same thing in narrative, there’s a clear difference between animals speaking in human language and people being able to understand animal language. Dr. Dolittle is probably the most famous example of the latter, but he’s hardly the first.
The ability to understand animals is a power King Solomon was sometimes said to have, presumably as an aspect of his wisdom.
And Siegfried was able to understand birds after drinking Fafnir’s blood.
There’s the interesting question as to whether animals really do talk in the sense that humans do, as we’re the species that has managed to complicate everything so much that constant talking has become necessary for us.
It’s an interesting conceit that I could theoretically translate what the cats say into English, but most of what I’ve read about the subject suggests they’re really only trying to convey a few key ideas, not converse about philosophy and current events. A lot of animal communication is done through body language as well, and to be fair Dr. Dolittle does learn that as well.
Still, it’s a common theme, particularly in children’s media, for animals to be regarded as carrying on conversations in their own languages. Sometimes animals are able to communicate quite easily with ANY other animals besides humans, while other times it’s only with other animals of the same sort. L. Frank Baum mostly goes in for the latter. While animals in Oz and some other fairylands can speak Ozish and know it to be different from their own means of communication (meaning it presumably isn’t some kind of universal translation spell at work), there are other cases in Baum’s fantasies of people being able to talk to animals in their own languages. In The Life and Adventures of Santa Claus, we’re told, “The language of the beasts became clear to little Claus; but he never could understand their sulky and morose tempers.” Queen Zixi of Ix is said to have learned “the language of the beasts and birds and reptiles” due to her witchcraft, although whether this means she learned them through study or used some kind of magic to gain the ability to communicate with them isn’t entirely clear.
The book that probably says the most about this subject is John Dough and the Cherub, in which the Elixir of Life gives John the ability to speak all human and animal languages. Indeed, the impression given is that a human could theoretically learn to speak the language of rabbits or bears just as they could learn a foreign human language. At one point, the bear Para Bruin says to John, “The Mifkets speak one language, and you and I another, and the Princess and Chick speak still another! And it is all very absurd, for the only language I can understand is my own.”
The gingerbread man has to act as translator. Baum doesn’t appear to have been entirely consistent with this, however, as Pittypat the Rabbit is later able to understand John’s conversation with Ali Dubh even though it isn’t in rabbit language, and after leaving the Island of the Mifkets Para has no trouble conversing with humans. In The Tin Woodman of Oz, Polychrome is transformed into a canary and the Tin Woodman an owl, and they’re able to converse in the bird language. And in Ruth Plumly Thompson’s Royal Book, the Comfortable Camel refers to the language camels speak amongst themselves as Camelia, and the Doubtful Dromedary later mentions Turkey and Donkey as other languages.