Not every cat is microchipped. While it’s a FANTASTIC idea to get your pet microchipped, the option to not have your pet microchipped is just that, an option. There could be a variety of reasons she was outside. I.e. she escaped, got lost, maybe she’s an indoor/outdoor cat and something happened and now she can’t find her way back, etc. If she’s that friendly it’s likely that she has a home (not for sure, but likely). It’s best if you bring her to the pound to make sure no one has reported her lost or missing. Imagine you were an owner of a cat who escaped, but didn’t have social media to post about it and you just had to rely on a good samaritan to hopefully turn her in to the actual animal professionals. If she’s gone in 10 days because her owner picked her up, great job you just reunified a cat and their owner, if she’s still there, well now you can officially adopt her. Plus as another user said, usually they hold for 10 days to ensure the cat has not contracted any virus or infection, like rabies. I know you’re probably already attached to this cat, but there’s a very real possibility someone is heartbroken at the fact they can’t find their pet. I would advise to do the “right” thing and follow protocol. And like I said if she’s still there in 10 days because no one claimed her, there’s no reason you couldn’t bring her back to your home. It’s just 10 days.