The term that's been around for centuries is "single cat lady". And even today, they learn too late that while they can whore themselves out to Chad and Pookie, they can't get a man to commit to them.
They're also known as Sheng nĂ¼, translated from Chinese as as 'leftover women'. Japanese women refer to Japanese women as "Christmas Cakes" if they're unmarried past 25.
For women who need to accept their fate and stop chasing relationships, the British have a term "AuntiePops", as in the aunts of GirlyPops (who are likely to become AuntiePops themselves).
There are several other terms such as PickMees and Pickmeeshas, Slump Busters, Tinderellas, "For Recreational Use Only", etc., and quite a few that would be unnacceptable on this forum.
And I have to confess, I don't get why men get so vicious about the cats. You yourself said that men get dogs for companionship, and you don't seem to feel that there is anything wrong with that.
But the answer is obvious: men do not criticize themselves for anything.
Femcel is another such term. There’s also the traditional Old Maid
How could I forget Spinster?