We definitely have his and her dogs. Part of this is because he adopted Cynder years before we met and she is 100% bonded to him, and when I purchased Abrams he was Mine and Michael was gone for 7 months of his puppyhood, so he really bonded with me as his primary human and not Michael. But even if it weren't for that, we still likely still "split up" the dog ownership.
Michael loves dogs and even if I wasn't with him he'd have a few. We both like a lot of the same traits, but I put a lot more stock on structural/genetic/temperament soundness whereas he is more of a, "Eh. It's a dog." And while we like a lot of the same temperamental traits, our aesthetics are completely different. He also only wants adult rescues, whereas I prefer puppies so I can shape them how I'd like vs trying to break habits that annoy me. (And since I prefer puppies, I'll likely only go through breeders if I am actively searching for a dog.)
I do more of the routine care with them, but that's just because it's, well, part of my routine. LOL He's more than happy to pitch in if I'm not able/ask. I feed, play, train, groom, and vet, he helps with the potty breaks, brushes/bathes when asked, feeds when I ask, and most importantly he plays/wrestles with Abrams and helps me tire him out.
I imagine we'll always have "his" and "her" dogs - my only issue is I like having two of "my" dogs and trying to fit that in with "his" dog(s) will take some finesse. And regardless, we will both have say on to whether or not the other is allowed to bring the dog/pup in to the family, because even if it isn't "our" dog, we still have to live with it and will be helping out with it.
(For what it's worth, the cat is definitely both of ours and not just one or the others. I feed her and clean her litterbox, but he dotes on her and plays with her and spoils her. If anything, I'd say she's more of his cat than mine. Haha)