Franklin: Practically everyone besides the title turtle. His friends include a bear named Bear, a rabbit named Rabbit, a beaver named Beaver, a fox named Fox, a snail named Snail etc. Apparently only turtles are creative enough to give their offspring names not taken after their respective species.
The main character in the short-lived cartoon series Family Dog did not have a name.
George of the Jungle: When he gets in a scrape, he makes his escape with the help of his friend, an ape named Ape.
Sonic Sat Am has Bunnie Rabbot. Justified in that "Bunnie" is an actual name.
There is also Cat, the cat. And Lupe, the Wolf.
And Muttski, Sonic's non-humanoid dog.
South Park: Cartman's cat is called Mr. Kitty. Ironically, Mr. Kitty is female.
Parodied with the holiday episode, "Woodland Critter Christmas", where this trope is combined with Buffy Speak. The critters (who also happen to be devil worshippers) are named as follows: Squirrely the squirrel, Rabbity the rabbit, Beavery the beaver, Beary the bear, Porcupiney the porcupine, Skunky the skunk, Foxy the fox, Deery the deer, Woodpeckery the woodpecker, Mousey the mouse, Racoony the raccoon, and Chickadee-y the chickadee. It makes sense considering the entire story was cooked up by Cartman at the end of the episode.
Every character in Word World. Which makes a certain amount of sense, since most objects in Word World are made of the letters that spell what it is.
Young Justice has the alien robot sphere named Sphere and the albino wolf named Wolf. Lampshaded when Conner (who named the above two) is introduced to a monkey named Monkey. He says that's a good name.
Terrytoons' Deputy Dawg had a character named Al E. Gator.