AITA for keeping this stray cat?

We had a seemingly stray cat follow us home a few days in a row while we were out for family walks. We spoke to neighbors who say this cat showed up a few weeks ago, it's always outside, friendly with everyone and no one has ever seen or heard of an owner.

We finally decide to feed him and take it to an animal hospital to check for a chip. He, of course, does not have one so we take him in. He was also not neutered, skinny, and had fleas. I call every local shelter and the police department to report the found cat. I post to lost pet Facebook groups. I talked to animal control and filed the found cat report. The shelter asked me to hold the cat, then after 7 days, if they owner hasn't come forward I could keep him and legally that would make the cat ours because no owners are looking for him. I was apprehensive to "foster" a cat because we have small children I knew would grow attached quickly. We waited 10 days instead to finally tell our daughter she could call him ours and accept our new furry family member.

Fast forward to today, two weeks since we've brought this cat into our home and reported him missing. Someone messaged me that the cat I posted is theirs. They say their cat has been missing for a week and they thought he just got hit by a car and we're so happy to see my post. Already this is weird to me because he's been in our possession longer than they were looking for him and he seemed to be lost before we even had him. I tell her I'm so sorry she's missing her cat but those descriptions don't line up and now we've already accepted this cat into our family and has been with us longer than her cat was missing.

Hours later, at 11pm she sends a bunch of pictures in a message I guess trying to prove this was in fact her cat. It's late at night, I hadn't answered, by midnight she posted me all over this lost pet page as having stolen her cat and refusing to give him back and ignoring her proof now. I wake up to over 150 comments telling her to get an angry mob to come to my house because the police won't do anything, posting my home address, my workplace, my photos with my children. They came for blood. I messaged this woman back to ask for any paperwork or proof the cat is hers. She has none. Says he's never been to the vet and couldn't afford to ever take him. Says she got him from family so no adoption paperwork, and they admitted to knowing he goes outside often but didn't neuter him or get him chipped or even a collar! I asked what she did to look for the cat. She says nothing but they were going to make flyers eventually. I tell her I'm very sorry but this isn't substantial proof that this cat belongs to her, I did my due diligence, and we are now attached to the cat and plan to keep it. She freaked out a bit and I blocked her.

Later tonight, she SHOWED UP AT OUR HOUSE KNOCKING OK OUR DOOR AND WOULDNT LEAVE. We had to have the police make her leave our property. We're keeping the cat.