The vast majority of cat and dog owners are criminally irresponsible and shouldn't own them.
Most cat owners let them outdoors and they roam around town without supervision, which not only exposes them to a lot of dangers from dogs, other cats, cars, disease and parasites, but they can also become a big nuisance when they invade people's homes or make so much noise people can't sleep at night. Not to mention that they can become feral. Cats are also responsible for killing 2.4 billion birds and 12.3 billion mammals every year in the lower fourty eight states alone, which makes them a massive biological hazard to any wildlife environment they are introduced in.
For dogs, a lot of them need constant exercise and mental stimulation that people that live in apartments and work a full time job simply cannot provide for. And when their owners let them roam free around the streets it's the same issue as cats, but there's also the fact that dogs present a much bigger threat to people on top of that, especially children and they also tend to form packs which only makes them more dangerous.
Most pure breeds are also very prone to health issues. Pugs' eyes can literally fall out of their sockets, they all suffer from hip dysplasia, spinal issues, respiratory issues and there's the fact that most females simply cannot conceive naturally, most pugs can only be delivered through a c-section. Some munchkin's legs are so short, most of them can't jump like any normal cat should be able to. And do I need to say anything about pitbulls? I could go on but there already is an entire
thread dedicated just to that. But despite
*all* these issues I've outlined, there's still a colossal demand for many of these breeds, no matter how unhealthy, deformed or plain dangerous they might be.