For different species, consider Disney's Robin Hood with Robin as a fox, Little John as a bear, Friar Tuck as a badger, etc.
Given that cats rule the Internet, it is probably not surprising that there are lots of cat examples. For stories told with animals check out these Myth-Folklore Storybooks: The Cat Gods and the Tales of Heraclaws (Greek myths with cats) and A Kingdom of Love in a Lion Pride (the story of Cupid and Psyche told with lions).
Here's the Biblical story of Esther with animated cat gifs: The Entire Story Of Purim, As Told By Cat GIFs. Here's just one of the gifs; meet King Ahasuerus:
And here's a poster that imagines a version of Star Wars with cats... it's Star Paws.