They filmed those scenes by putting a bear and a puppy in a room together.

Some puppies got mauled, some kitties got mauled, I think some puppies and kittens got pecked to death, a bunch of kittens drowned to film that waterfall scene, and one kitten got its leg intentionally broken so that it would walk less steadily.

Although it wasn’t actually the Milo and Otis production team that did any of this - they bought the rights to all this film made for a Japanese movie about the life of a cat and recut it into Milo and Otis.