Everyone seems to have forgotten about all the live action movies Disney has made. There are movies such as "The Shaggy Dog" and "The Shaggy DA." The plot of both are based on the premise that a man and his pet dog have switched consciousnesses. Here's a list where an animal was one of the starring roles.
"Old Yeller"
"Nikki, Wild Dog Of The North"
"Greyfriars Bobby: The True Story Of A Dog"
"Big Red"
"The Legend Of Lobo"
"Miracle Of The White Stallions"
"Savage Sam"
"The Incredible Journey"
"A Tiger Walks"
"The Ugly Dachshund"
"Monkeys, Go Home"
"Charlie, The Lonesome Cougar"
"The Horse In The Gray Flannel Suit"
"King Of The Grizzlies"
"The Biscuit Eater"
"Run, Cougar, Run"
"Chandar, The Black Leopard of Ceylon"
"Mustang"
There are more, but you get the idea.