Tama from Hayate the Combat Butler. He speaks perfect Japanese, can walk on two legs, has a degree in economics and is a qualified boiler room operator. Hayate is the only person who knows he can do it, however; to the rest of the cast he is just a lovable, oafish and above-average intelligent tiger (or 'big cat'). He's also spoken in front of Wataru. When confronted about speaking in front of others he says something about not wanting to shatter the girls' (Maria and Nagi) dreams.
In Studio Ghibli's Porco Rosso (The Crimson Pig) the main protagonist is a pig, or more exactly a "pig-headed" human as he has the complete anatomy of a stout person except for the head. He is the only one of his type and lives among a human society who, while aware of his difference, don't find it bizarre and sometimes openly point it out. It's stated that he somehow became a Baleful Polymorph, given that there are pictures (and a flashback) showing him in human form.
Meowth of the Team Rocket trio taught himself "human talk" and how to stand on his hind legs, all for the love of a female Meowth named Meowzie who said that a human could do more for her than anything he could (as in, the food, housing and adoration), only to be rejected as now she considers him a freak.
In one manga adaptation, a story arc centers around the kitten of the very same Meowzie; she's found in a Poké Ball the gang purchases (unable to get captured herself, the mother pops her kitten into an unsold ball to ensure that whoever finds her will give her a good home). When Team Rocket, and thus Meowth, come around making trouble, Meowth meets this kitten and instantly sees her mother in her face and shining coin. Then she tells him it's creepy how he acts like he's a human. Oh heartbreak.
Princess Tutu's Mr. Cat looks exactly as his name implies, and occasionally meows and cleans himself with his paws, but is otherwise a marriage-obsessed ballet instructor. While he is the most prominent one and has the most screen time, there are other guest characters that also fall under this trope.
There's a couple examples in Dragon Ball Z, which stick out like sore thumbs among the overwhelmingly human/space alien cast.
Oolong, an anthropomorphic pig.
Puar probably also counts. They only stick out if you never saw or read the original Dragon Ball, which was a much more light-hearted series and almost overflowing with Funny Animals as secondary and tertiary characters. Dragon Ball was originally based on Journey to the West, which had Zhu Bajie (known as "Cho Hakkai" in Japan and "Pig" or "Pigsy" in most English translations) and Sun Wukong (better known by his Japanese name "Son Goku") as two of the main characters-Oolong and Goku, only in their original forms.
Tony Tony Chopper, a gullible reindeer doctor. Justified in that he ate the Hito-hito no mi (Human-human fruit), which gave him human intelligence; before that, he was an ordinary (albeit blue-nosed) reindeer.
Bepo, an apologetic, self-conscious polar bear. Whenever he's aware of people who are surprised at the fact that he can speak, he'll get depressed, hold his head down in shame, and apologize to them.
When Pappagu the starfish was young, he mixed up the word hitode (the Japanese word for "starfish") with hito (a Japanese word for "human"), so he thought he was human. By the time he realized he was a starfish, he had already learned how to speak like a human. He has also created the successful "Criminal" clothing line.
Pekoms, one of Big Mam's subordinates is a lion, with little eyes (which he counsel behind sunglasses) a formal attire and excellent fighting skills.
Shirokuma Cafe is pretty much about this, as well as their interactions with humans.
Inverted with Mr. Shoebill from Episode 8. He does not talk or act anthropomorphosized, but he is still sentient as the Funny Animal characters and is the editor in chief of a local food magazine.
Kermit (and other frogs), Piggy (and other pigs), Fozzie, Rowlf, and assorted other characters on The Muppet Show. (Scooter, Bunsen, the band and the Whatnots are probably meant to be humanish, Statler, Waldorf and the Swedish Chef are definitely Muppet humans, and Gonzo is ... whatever.)
Otto, Sarge's dog in Beetle Bailey, is another example of a non-talking Funny Animal. When first introduced in the strip he was more of a regular dog, but over time Mort Walker began having him walk upright, don a full uniform, etc.
The Far Side has funny animals of all kinds, from cows to insects to amoebas (the Rule of Funny superseding the fact that protozoans are not animals).
Hobbes from Calvin And Hobbes. Hobbes is really a mix of Funny Animal and Talking Animal; he walks on two legs and sleds, plays baseball, and talks with Calvin sometimes and walks on four legs, pounces, ands runs around friskily other times.
Beyond Good And Evil has these for most of the NPC's: Pig-men, shark-men, walrus-men, etc. Taking pictures of them with the camera names their species in the form of "[Species] Sapien."
Brutal: Paws of Fury, a game that came out during the 16-bit fighting game boom, has a cast entirely made up of funny animals (most of them Half Dressed Cartoon Animals)
Bug. Most of the arthropod enemies in the game aren't, though.
Cave Story. The Mimigas from are rabbit-people. Other than subsisting on flowers (and not wearing pants) they don't act any different than their human neighbors. (And at least two of them are humans who were magically transformed.)
Crash Bandicoot, well, to an extent, with Ripper Roo and Tiny Tiger who averts this somewhat.
Tail Concerto and Solatorobo are set in a world of floating continents in the sky, populated by dog-people and cat-people. While they mostly act like humans, the dogs are prone to chewing on bones and their national pastime seems to be frisbee-cathing.
In Everyday Heroes, Summer's classmate is has a cow-like head and tail. She and her father are actually from another planet, yet no one comments on their appearance. Word Of God has it that other alien races fall under this trope (for example, the "Dogs Of War").
In Harkovastevery character is a talking animal person, with each nation being made up of a particular animal. The only possible exception is the Nameless, whose species is indeterminate at the moment due to their all covering armour and helmets.
Porky Pig and (sometimes) Daffy Duck are rare Looney Tunes examples. (Porky is probably like this because he was created earlier, in an era when Disney had made Funny Animals universal; Daffy probably learned how to drive and wear clothes from hanging around Porky.)
Courage the Cowardly Dog. While everyone acknowledges that Courage is a dog and usually treat him as such, other Funny Animal characters get by with no reference at all to their species. Courage's own characterization as a dog slips sometimes. Often the alien/evil entity/villain of the day (who is sometimes also a talking animal) will treat Courage on the same level as the humans. Often by trying to kill them.
The penguins in Happy Feet are an odd case; at times they uncomfortably straddle the line between penguins who can talk and penguin-people.
Brian Griffin of Family Guy usually falls squarely into this category, albeit with occasional Talking Animal moments (particularly in the first run of the series, where they were more frequent).
The chickens in Chicken Run are difficult to categorise—the humans treat them just like any other chickens, but most of them wear at least one article of clothing and the stupidest one is capable of knitting.
All members of the London Clan in Gargoyles resemble humanoid animals with feathered wings. The Mutates are a genetically-engineered, chimeric version.
I Am Not An Animal. The animals were genetically engineered to talk and were raised with celebrity magazines photoshopped so half the people had animal heads. They escaped and didn't realise that animals aren't supposed to talk.
Regular Show both plays this straight and averts it with its main characters. Mordecai the bluejay might as well be a human, seeing as he doesn't eat like a bird, fly or do anything else birdlike. Rigby the raccoon, on the other hand, is shown to dig through trash and run on all fours, but otherwise acts almost as humanlike. Lampshaded in this conversation:
Mordecai: Dude, don't dig in the trash. It's not natural.
Rigby: You're not natural!
The entire cast of The Great Mouse Detective could be replaced by humans and it wouldn't make any difference at all. Well, except for Toby who is very much a dog.
Strangely Popeye The Sailor encountered one of these in "The Hungry Goat". The short overall felt like more of a Tex Avery cartoon than a Popeye cartoon.
Funny animals are featured prominently in "Popeye The Sailor" and "I Eats My Spinach". Justified, as Fleischer Studios was known to use these in the Betty Boop cartoons, where the Popeye series branched out of.
Although some of them dip near-ish the Talking Animal end of the spectrum sometimes, the cast of Tuff Puppy is essentially this. Although they retain some animal like mannerisms (Dudley, a dog, has a superhuman sense of smell and an unfortunate habit of chewing his own butt, and his partner Kitty loses all self-control and star whenever she sees a mouse or a ball of string
Chip N Dale Rescue Rangers is straddles the line between this and Talking Animal, with a healthy dose of Furry Confusion for good measure. The main cast usually falls squarely into Funny Animal territory, though at times might slip into Talking Animal (Chip and Dale more than the others, as they will sometimes run on all fours like actual chipmunks. Monterey and Gadget almost never behave like actual mice). Most other rodents in the series are also portrayed fairly consistently as Funny Animal. Cats generally tread both sides, while dogs and almost everything else tend to be Talking Animal exclusively. Furry Confusion occurs with pretty much everything, particularly fish and incidental or background characters.
The cast of Kung Fu Panda. In particular, the characters use their animal attributes in their fighting styles, like Crane's wings or Po's body fat. Tigress and Tai Lung come the closest to crossing over into Petting Zoo People, but they are still distinctly feline.
Older Than Dirt: The Ancient Egyptians apparently liked funny animals. The story of The Mouse As Vizier features talking animals who have a very human political system, and several papyri depict animals such as mice, cats, hyenas, antelope, crocodiles, donkeys, monkeys, and lions playing board games, using weapons, drinking out of goblets, and playing musical instruments. Except for standing on their hind legs, they aren't anthropomorphic at all. And they completely ignore natural predator-prey relationships. They even herd livestock and ride chariots pulled by normal animals.
A fancy Sumerian lyre, dated to c. 2600 BCE, features inlay scenes that depict funny animals. A bear, jackal, and donkey play music, while a lion and an antelope serve beverages. Except for standing on their hind legs and having hands, they look like normal animals.