Forget even costs, most people with pets, especially dogs, shouldn't own pets. They have no understanding of a dog's needs, instincts, how to train them, or why, and generally are entitled to the extreme. A dog becomes almost like a weapon under their care. And many of those people are not on low-income, by a long shot.
Sure, you should make sure you can take care of your pet's medical needs, but equally make sure you do your research, understand what kind of pet you're adopting, and that you can take care of their emotional and psychological needs, spending time with them, knowing how they think, walking and exercising them - it's a huge responsibility towards a creature who can't grow up and leave home when they get fed up with the abuse. I'm not sure that's classist, it's just compassionate. No one has to own a pet, or at least, very few do. Those with service animals, I think, get the costs covered.
Don't even let's get started on can people do these same things for their actual human children...