Eventually, the animal fable developed in so many directions that it stopped being a Thing. There’s The Country Bunny and the Little Gold Shoes (1939, a surprisingly woke book for its time, by the author of the play that Porgie and Bess is based on); Rabbit Hill (1944); The Rescuers (1959); A Cricket in Times Square (1960); Mrs. Frisby and the Rats of NIMH (1971); and Watership Down (1972). All of these break out of the animal fable in one way or another, developing new forms.
Perhaps closest to the original genre, in mid-century, were the books of Eve Titus, including the Anatole series (1956 onward) and the Basil of Baker Street series (1958 onward).
While all kinds of sentient-animal stories have been published in more recent years, there is nothing quite like the absurdist throwback, Mr. and Mrs. Bunny–Detectives Extraordinaire (2012). It starts in a fairly realistic way, with a girl named Madeline being raised by hippie artist parents on Hornby Island in Puget Sound. But then her parents get kidnapped by, um, foxes, and she has to enlist the help of bunnies. Bunnies who wear fedoras.