**Barenaked Ladies **- “Tiny Little Song,” all from the POV of a cow.
Sam the Sham & The Pharaohs - Li’l Red Riding Hood
“Terrapin” by Syd Barrett (Sung not by a terrapin, oddly enough, but by a fish.)
Genesis’ “A Trick of the Tail” is sung by a mythological beast.
“I Am Your Pussy” by Gong – yes, it’s sung by a cat, double entendres notwithstanding.
“Wet Dream” by Kelly Monteith–a fish.
I have to admit I haven’t heard of any of the other songs listed in this thread, except this one–and it’s really an inspired choice. 
Thanks to everyone thus far.
Bright Eyes, from the Watership Down film, is from a rabbit’s POV.
“I Wanna Be Like You”, sung to Mowgli by King Louie the Ape.
The Weakerthans: Plea from a Cat Named Virtue
Peter, Paul and Mary-Leatherwing Bat
Isn’t the entire soundtrack, except “My Own Home”, sung by the animals?
Jonathan Coulton’s Space Doggity, sung partially from Laika’s POV.
Sandra Boynton has put out 3 CDs where many of the songs are sung by “animals.” Notable is “Penguin Lament” by 5 for Fighting.
XTC’s “Fly on the Wall”: “I am the fly upon the wall/My prying eyes are looking through your bottom drawer/I just came flying through your door…”
Belew also wrote the lyrics for the King Crimson song “Dinosaur,” from the P.O.V. of a fossilized dino: “…what an easy prey I must have been/standing in the sun, idiot savant/something like a monument/I’m a dinosaur, somebody is digging my bones…”.
You’re right, although “I Wanna Be Like You” is specifically about being an animal and not a human.
Kate Bush’s song “Ne T’Enfuis Pas” is about a cat stalking a bird.
Puff the Magic Dragon.
StG
Flanders and Swann’s “The Elephant” and “The Sloth” are entirely from the title critter’s point of view.
The song is sung in the third person by an unknown narrator, not by the dragon himself.
From a sort-of animal’s perspective: “Furr”, by Blitzen Trapper. (Great, great song.)