I love your vet.
Abigail isn't a bad name either.
I'd also have such mixed feelings if I were in your shoes. I'd be very troubled by the thought that someone might be frantic and missing their beloved pet, but, at the same time, it would be difficult not to start getting attached to such a sweet and appealing dog.
Not everyone does make an effort to find the owner when a dog without ID follows them home or suddenly turns up on their doorstep. Many years ago, I had a neighbor who found a dog sleeping on her front porch one morning. It was a cute small dog, a well cared for mini schnauzer, who had obviously wandered away from her home, but neither I nor my neighbor, nor the other dog owners in the immediate area, could remember seeing the dog walked by anyone.
My neighbor said she'd wait for someone to come looking for the dog, and she began referring to her as "a freebie" which got me very angry and I told her that, if she didn't try to find the owner, I would. I was really appalled by her selfishness, and pretty dumbfounded that a dog owner would be so cavalier about wanting to hold onto a dog that was obviously someone's pet--this was before the era of microchips, but the dog was wearing a collar without tags, and she obviously hadn't been on her own long at all. I finally convinced my neighbor to call the police department because we have a small local police force that does often get involved when people are searching for missing pets and, sure enough, they gave her the name and address of the people who were looking for this dog. But, I really think that if I hadn't pressured my neighbor, she would have just held onto that dog. I have a hard time understanding people like that.
I've known people who did not keep their dog's collar on in the house, they kept it attached to the leash and just put it on when they took the dog out. I've known others who never put a collar on the dog because they always walked the dog with a harness. If those dogs suddenly bolted out of the door they'd lack a collar and tags. So, I can think of possible reasons why a dog might be wandering around without a collar, in addition to the collar possibly having opened or broken while the dog was out being walked or even tied in the backyard. Those things are all possibilities in Snowface/Abigail's case.
I'd call all the local shelters to see of anyone is searching for her, and if you think the police could be helpful, as they are where I live, I'd call them too. But, there are limits to what you can do. The owner would have to be making similar efforts to find her. I think it's unlikely that someone would just abandon a three year old pure bred. well behaved, dog, but you never know.
I think if an owner is going to materialize, through some channel, that it would happen relatively soon. Meanwhile, it sounds as though everyone involved is enjoying each other's company, and you are certainly taking excellent care of her. If, heaven forbid, my dog were ever lost, I'd fervently hope that she would be cared for in just the kind and loving way you are doing with this dog.