This topic is really interesting to me, since I do love to pick apart brain chemistry and figure out why humans and critters do the things they do, and I do have a few theories for this...
Like SillyKai said, a lot of folks may feel connected with their pets in some way. With dogs and cats being the most popular pets out there, I’m sure a lot of folks have had decent exposure to these creatures. Folks may see their pets as family, as a solace, as a safe place, and this mindset could strengthen if they’ve struggled with forming close bonds with humans...something that many neurodivergent folks struggle with, and I have noticed a great deal of therians being ND. Pets tend to love their owners unconditionally, unlike humans, and that’s where a sense of belonging could come in. Also the fact that pets don’t need to bond through speech could be a big contributor, since one of the biggest things that NDs struggle with is communication. When it comes to animals, as long as another animal isn’t a predator, prey, or trying to harm them for whatever other reason, then they’re fine with just...existing near each other. They don’t even have to speak
My other theory is how much we tend to observe in dogs and cats compared to other animals. When we’re exposed to a certain animal for a long enough period of time, we’ll notice little things in their behavior and think “huh, that’s a lot like me actually...”, and with dogs and cats being so common, of course folks will be observing them the most. Since we’re so used to any other kind of animal being territorial, asocial, and...a bit dim to be frank (since we don’t learn about them as often), it’s hard for people to relate to them. But both dogs and cats have the potential to show a more cuddly, social, and intelligent side to them when we bond with them and train them, making them more relatable. However, there could be folks who do feel a bit territorial and asocial (like yours truly), and they tend to gravitate towards more wild animals. They’re not quite as common, however, which is why dogs and cats are the more common theriotypes.
But, eh, those could be incorrect. That’s just what I’ve observed from watching the therian community and my theories are greatly based off of my own experiences. It could honestly be because of a whole slew of reasons, but who knows?