Ruffio & Watts
Now I know that it isn't entirely the same as taking human portraits because you can't really "pose" a cat or a dog to help accentuate their best features. For example, pets won't cooperate if you ask them to tilt their chin at a 90-degree angle while turning their head slightly to the right to hide that double-chin. Having a moving, living, breathing subject is very different than still-life. They don't always cooperate. Sometimes they break out into wrestling.
But the practice does accentuate your powers of observation with a live subject and the different dynamics that could include. How the subject interacts with its environment and how environment affects the visual mood. How a light source from the left casts a shadow on the right, adding depth and contrast. How just even slight changes in expression can entirely change the feel of the picture. Those are just a few examples, of course, but you get what I mean.
Two Kitty Paws
The best part? If session doesn't go well or if the practice portaits don't turn out, the "client" won't care. And, if your subject is your own pet, you can practice, practice, practice until you start liking the photos that you see!