It would be cute if we could have cats and dogs for pets. You could adopt one from a store in some area of the game for an "adoption fee". Then you could buy it things (treats, yarn balls, sticks for fetch, food bowls, etc.) and keep it in your house. You could name the pet. Playing with it would raise your mood. Feeding it would raise your energy. You could feed or play with it as much as you like, but only earn points for it once per game day.
Other possibilities: If you buy a carrier case, you can transport your pet with you in your inventory. You can take your pet to visit other people's homes.