In my WIP, my heroine's father came and went while she was growing up. She never really attached well to him because she was protecting herself from the inevitable hurt when he left. So instead of giving her a dog or cat that she could grow too attached to, I gave her a goldfish!
My heroine in Burning Touch, an erotic romantic suspense novel, Devon, the heroine, rescues cats because her folks died in a car crash and she felt helpless to rescue them.
In Tropic of Trouble, also an erotic romantic suspense, corrections Sergeant Jason Jones is a tough guy. And he has a tough guy dog - a German Sheppard.
I often use a pet's name to convey something about their owner. A modern day witch in one of my works in progress has a cat (which isn't the cliche black), named Valiente. A reader might have to dig a little to figure that one out. Doreen Valiente was a British witch who lobbied to repeal the law that made witchcraft a crime in England.
Our choices in pets are just one of the many facets that make characters and people who they are. I've posted a picture of my sweet kitty below. She's the best, most sensitive cat we've ever had and she is also my writing buddy.