There are a lot of connection to cats and women, as well as dogs and men. You ain’t wrong, OP.
My guess is that, like @Zaku said, cats are perceived as elegant, quiet, and clean (and territorial, which is quite similar to being…..catty. Just like women) while dogs are much louder, agressive, and social, which is more stereotypical of men. Not that women ain’t social, but they were usually stuck in the house, while men spent much time everywhere else.
Also, cats usually live around the household, where women (stereotypically) are. Dogs are used for hunting, a male activity. If they’re at home, they are guarding the house. Guardians, in our societies, are men.
Also#2, cat is a name for a species and a female member of it. Dog is name for the species and a male member of it. Male cats and female dogs have specific names. This happens in other languages too.
Also#3, cats were usually connected to witchcraft, which is almost exclusively connected with women.
Also#4, cats, like @YARNLADY mentioned, are largely female. Male cats are less common. Dogs are largely male, and female dogs are less common.
Just some thoughts.