Seriously, did Kenneth Graham even know if he was writing about Funny Animals or Talking Animals? The opening scenes suggest more the latter than the former: yes, Ratty has a boat, but he lives in the riverbed, and compliments Mole on his "velvet smoking jacket", which is clearly Mole's actual pelt. Then we learn Toad lives in a big house and it gets a bit confused. Then it turns out Toad can drive a car, be tried and convicted as a human, and successfully disguise himself as a washerwoman. And meanwhile we're back to the other animals living in holes and being completely disconnected from the human world. What's going on? By the end of it all, I don't even know how large these animals are meant to be, let alone whether they usually wear clothes!
Congratulations, you just came across the Anthropomorphic Shift. It's not that much of a big deal, really, though I have to agree it gets a little confusing. Just remember that everything is an allegory, and you'll get along fine.
Kenneth Grahame himself addressed it: when asked by a fan how Toad could drive the train, he answered that "Toad was train-sized and train was toad-sized".
I always thought it was intended as satire, the characters representing various types of Edwardian gentleman ... It's no weirder than Hey, Arnold!
Since he wasn't reading TV Tropes, he didn't have to make his characters fit only into one or the other of those categories...
While we're at it, it's also a deliberate Anachronism Stew. As Toad goes into the jail we time-travel from the turn of the century into the Middle Ages, walking past "men-at-arms" and "ancient warders" with halberds and a room with racks and thumbscrews, and by the time we've stepped into the "grimmest dungeon... in the heart of the innermost keep," the Edwardian police sergeant is starting to say things like "Oddsbodikins!" and "a murrain on both of them!" It's hilarious.
A. A. Milne's theory was that the animals were actually fairies, so they could do whatever they liked.
I love that theory. When The Call comes tell it that it's a fairy. Please?