Three words: “Crazy cat lady.”
Four words: “Crazy old cat lady.”
The original phrase reveals three pernicious biases: mental illness stigma, singlism, and misogyny. That’s a bigotry trifecta and if you stir ageism into the mix, you’ve hit the jackpot.
A single woman with a cat is usually assumed to be starving for human companionship, projecting that onto an “unaffectionate” animal assumed to be standoffish, too smart for its own good, and mysterious (much like women often are). Dogs are assumed to be friendly, outgoing, happy, and perfect companionship … for men.
Therefore, it can be surmised that a woman with a dog benefits from indirect male privilege, just like women with “nontraditional” (read: male) occupations, hobbies, and interests.