I love animals. I was a quirky and somewhat introverted kid, and I took a great deal of comfort from my relationships with my family's companion animals. My fascination with animals has continued into my adult life, and in addition to my husband and myself, our household includes three dogs, three cats and a gopher snake.
I also read voraciously as a child, and many of my favorite books had plots that revolved around animals. Some of these were fantasy stories, some were clearly rooted in the real world, and quite a few were somewhere in between. Animal-focused stories are less common in adult fiction, but they definitely exist. In general, books which revolve around animals seem to fall into five broad categories:
1. Stories where the main character(s) are humans, but their relationship with an animal or animals is the major focus of the plot. The reader only has access the human character's thoughts and perceptions via first person, limited third or omniscient narrative. The story doesn't get inside the animals' heads. Examples of this would be Old Yeller, The Black Stallion, Mr. Popper's Penguins etc.
2. Stories where the story at least partially revolves around the animal's experiences, but the author does not put human thoughts or words into the animal's head (though they may, via omniscient narration, imply that the animal has a high level of intelligence and prescience). Examples of these would be: Lassie Come Home, Terhune's Lad of Sunnybank books, and many of Earnest Thompson Seton's stories about animals (which always left me in tears).
3. Stories where the animals speak to one another with words, and have very sophisticated thought processes, but where the animals are still very animal like in some ways. Watership Down, Bambi and Black Beauty are examples of these kinds of stories. These animals generally cannot speak to humans, and they may in fact be victimized by humans in the story.
4. Stories where the animal characters are really very much humans in fur suits. In these stories, the animals not only talk to one another, but they may wear clothes and live in houses (or at the very least, if they live in burrows in the woods, they may have possessions like furniture). These are often stories aimed at younger children. Examples include: Lawson's Rabbit Hill books, Beatrix Potter's stories, Richard Scarry's stories, Beverly Cleary's The Mouse and the Motorcycle books and Eve Titus's books about Anatole the mouse.
5. Fantasy or Science Fiction Stories that have "wise beast" characters of one kind or another. By "wise beast" I mean animal characters who, because of magic or technology, have human levels of intelligence and can communicate with humans, but they are still not at all like humans. Examples of these would be: The Chronicles of Narnia, Jane Linksjold's Firestarter books, Robin Hobbs's Farseer Trilogy and David Brin's Uplift Wars trilogy.
So why do people, and most especially kids, love animal stories so much? One reason, I suspect, is that animals are forced to adapt to a world they didn't make and often required to live by rules they don't understand. Kids (and many adults) can certainly relate to this.
Another is that animals allow us to tell familiar stories and revisit familiar themes in ways that feel fresh. Watership Downs is hero's journey tale, but the fact that the characters are rabbits makes it an intriguing read to many people.
Animal characters also bypass many of the issues that come from selecting human characters of a particular ethnicity, or even gender, as protagonists. The late Richard Scary said that he used talking animals in his books because he believed that would allow kids of all ethnic backgrounds to relate to his characters equally. I don't know if this is true or not, but no one can deny that his whimsical books have been popular with many generations of children.
But I think another reason people of all ages continue to enjoy animal stories is not so far off the reason so many people love speculative fiction: Stories about beings who are not us but whom we can still relate to are fascinating.