So, there seem to be a few ideas popping up again and again in this thread.
1) First and foremost, cats are one of the most popular pets in America, and in fact around the world, and anyone who thinks it's somehow "wrong" for a guy to own one is a judgmental moron/asshole, and not really worth your attention.
2) Cats are seen as feminine, clean, low-maintenance, arrogant in a way that is somehow charming, somewhat lazy, and adorable.
3) Dogs are seen as masculine, relatively high-maintenance, loyal, hard-working (sometimes), and fun to play with.
I really don't see how anyone can claim that one is objectively better than the other. They're different kinds of pets, and they speak to different pet-owner personalities. And this doesn't even account for the fact that there are several different breeds of both cats and dogs, each with their own subtle, and sometimes not-so-subtle differences in personality and level of dependence.
Personally, I think it boils down to "dogs are more fun, but cats are less needy." That's kind of a gross oversimplification, but again, it's just my opinion.
I'd actually kind of like to examine the social impact of owning a cat or a dog, but I don't really have the time to dedicate to proper research. I think there's problems with the simple statement of "cats are feminine, dogs are masculine," and what it says about our assumptions of gender roles. I guess this means women are supposed to be demure, aloof, self-reliant and independent, cute, and generally prefer to coax someone into doing something for them than doing it themselves. Meanwhile, men are supposedly comparatively less intelligent but very loyal, hard-working, love to goof around just for fun, but you kind of have to train them to get them to do what you want. Some of that doesn't sound so bad, but some of that sounds kind of offensive.
Anyway, although "a guy who owns a cat is clearly gay" is an asinine statement, I can understand why a woman might prefer a man who owns a dog. Since the general perception is that cats are so much easier to care for, if a guy owns a dog, especially a well-behaved one, that's a more useful package of information for a woman. It means he's nurturing, patient, and kind of a goof. Let's face it, you have to have some appreciation for silliness to love your dog, and all the stupid, hilarious shit it does. Meanwhile, if you own a cat, all she can really tell from that is that you like company, but not too much. It's just less useful as a gauge for what kind of person you are. It's almost more useful as a gauge for the kind of person you're not than the kind of person you are.
Besides, how many people in this thread have said that they prefer cats because they're too lazy to keep up with a dog? That's fine, but keep in mind laziness generally isn't seen as an attractive quality by anyone, regardless of gender or sexual orientation.