I'm not sure it's about the animals so much as it is about the safety of the rules. Wolves will always be greedy, foxes will always be cunning, and the slow tortoise wins. That repetition builds a world where actions have predictable consequences, which is a huge comfort when you're small and the real world feels chaotic. They learn the patterns of storytelling—the set-up, the misstep, the lesson—and that scaffold lets their own imaginations start building on top of it. My kid will take the basics from 'The Three Little Pigs' and then spend an hour narrating a story where the wolf tries to sell them insurance instead. The fable gave him the pieces; his brain did the rest.
Some people argue modern kids' media is more complex, and it is, but there's a raw, almost primal simplicity to animal tales that cuts through the noise. You don't need elaborate backstories for the characters; a lion is a lion. That leaves so much space for a child to fill in the surroundings, the voices, the smells of the forest. It's low-detail, high-impact worldbuilding, perfect for a developing mind that's still learning how to connect narrative dots.