Seriously, did Kenneth Grahame 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 the 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?
Why is it that the instant Toad gets back to Toad Hall, the police seem to forget about him? Why do they never come for him and take him back to prison? Even if they didn't check, word would have got around about Toad's return and the weasel fight. In fact, why did no one go to the police about the weasels invading?
This has bugged me since I was a child. In "Dulce Domum", the Mole feels nostalgic about his old home (he had been living with the Rat for some time) and both animals make an off-handed visit to the Mole's previous residence. They are tired and hungry, and the Mole is distressed about not having any food around. After searching, they find a little provisions: "a tin of sardines—a box of captainís biscuits, nearly full—and a German sausage encased in silver paper." The Mole feels guilty about the measly dinner but the Rat tries to cheer him up. Then a bunch of kids singing carols show up, and the Mole is further distressed because they have nothing to give them. So the Rat promptly produces some money and sends one of the children to buy a full banquet for them all. My question is: if the Rat had money enough on him to treat a party of children, why he didn't go earlier to buy food for the Mole and himself? The Mole was miserable to have taken his friend to his empty house, and no cheering could hide the fact that there was barely enough cold food for the two of them. Couldn't the Rat just have said: "You, old chap, stay here in your lovely house, and light up a nice fire while I go get some dinner"?
Rat didn't want to leave Mole alone, since Mole was feeling pretty down on himself and his house. In addition, when the carol singers arrived he asked if the shops were still open at that time of night. This shows he doesn't know the area, so there was no way that he could have fetched the supplies; but he could send someone who did.