I woke up this morning full of hope that someone, somewhere, would find or develop a new form of misogyny, because face facts, we don't have enough right now.
Superficially, the analogy of cats representing the femine and dogs visa versa stands. Sure. But the differences between them are more profound. Dogs were domesticated well before cats, perhaps 10,000 years sooner. Cats are wilder. That's why of the thousands of pet abandoned every years dogs are more likely to starve while cats have a slightly better change living feral.
Dogs, in short, depend on us to a far greater degree. And socially, they are more similar to humans than cats.
These are reasons, aside from sexism, some people prefere dogs over cats. (In full disclosure, I'm partial to both, with perhaps a slight bias towards dogs. But we've usually had both in our household).
Finally, the assumption of bias in favor of dogs over cats is largely Eurocentric. The People's Republic of China only recently outlawed the dog meat trade there. But it's still big—e.g., the annual Yulin Lychee & Dog Meat festival. Street vendors in SE Asia—Hanoi, Bangkok, or the Philippines, or S. Korea—boil dogs alive because the flow of adrenaline from their final agony is thought to enhance the quality of the meat.
Cats are sometimes victimized this way as well. But they're harder to handle & don't have as much meat. Also, as the Chinese people learned after Mao declared cats bourgeois, their absence encourages the proliferation of vermin.
Dogs are gradually gaining ground in Asia to assume their proper role as working & companion animals. But they have a long way to go.