Help regarding a stray

Hi everyone! I was recently befriended by a lovely little long haired tabby stray. The poor little guy was riddled with fleas and undernourished, we fed him a few times and he kept coming back and would cry outside the window. We invited him in for the first time 2 days ago and he came in VERY willingly. I have a 2 and a 1 year old so was concerned with how he would react but he rubs up against their legs and lets them pet him and cuddle him. He is the most affectionate cat I have ever come across, he was curled up on my lap most of the day yesterday and even slept next to me last night. Anyway, we treated his fleas as soon as he came in and set up a litter tray, food and bed etc and my partner took him to the vets. He isn't microchipped and the vet estimated he was around 5 months. At this point I am now unsure what to do? We have called all of the local vets and no one has reported a cat of his description missing. We would happily keep him as he is absolutely wonderful but he seems to want to go out sometimes and I'm worried we're doing the wrong thing forcing him to stay in suddenly? Are we even legally allowed to keep him? On a side note his stools have also been very loose (I seem to remember our first kitten had loose stools at first so I wonder if it's the change in diet etc), should I be worried?? Sorry for all the questions we are just keen to find out the best thing to do by this lovely little kitty!!!

This is how lovely he is, a picture of him and my daughter

Thank you so much for rescuing this kitten.
A couple of thoughts: At the vets was he tested for worms? FIV/FeLK? Worms can cause smelly poop and a change in diet can cause loose poop. Impossible to say what he was eating on the streets. I don't much about canned food (I feed raw), but I'd look for a limited ingredient wet food, maybe with chicken. Avoid fish for the time being, and dry food can cause really smelly poop in sensitive cats.
That he was malnourished means to me he'd been on his own for a while. You can put up flyers and post in PetFinding "found" cat, but I think you're safe in keeping him. Laws vary from city/county/state regarding 'ownership' of property; cats are in almost all jurisdictions still considered property, like a bicycle.
If the cat is not neutered, he should be as soon as possible; otherwise, he will REALLY REALLY want out in order to mate, and may start spraying.

There are no laws I've ever heard that would restrict you from keeping a stray cat you took in. I think you should keep him even if the "owner" shows up--the owner was obviously VERY neglectful or this cat wouldn't have been malnourished and riddled with fleas. In fact, if I were you, I wouldn't even look for his "owner."

At my previous job, a neighbor's cat kept showing up at the back door patio area of our building. It was very malnourished. I bought  a large bad of cat food and began feeding her, and other co-workers started to feed her and even play with her, too. A co-worker picked her up one day and took her home. The vet said she was very malnourished, a "dumpster cat," and had worms and flea infestations, etc. We learned who "owned" this cat, and the family was informed that the cat had been taken in and was being cared for. The family did not care because they had no interest in caring for the cat. The cat is very healthy, happy, and loved in her new home. The point is that if someone loved the cat you are now caring for, it would not have needed to have been rescued by you.

If you cat isn't crying non-stop and clawing at the windows and doors to get out, then I wouldn't let him out again, ever. In a few months he will forget his former life and be content to sit in the windows and look outside without needing to go outside. Five months old is the perfect time to get him neutered--I wouldn't wait more than another month because male cats go though puberty at around 6 months.

Thank you so much for your responses. He wasn't tested for worms etc but next week we are getting him a full health check, vaccinations etc and we aim to have him neutered within a fortnight.

Well the local vets have his information but honestly if someone has lost him it would have been some time ago (unless he was seriously neglected already) so I assumed they would have reported him missing with any local vets/websites etc. aside from checking internet sites I don't think we will make too much of an effort to find a previous owners as he was in such a bad way and he is already very loved.

He seems super content 99% of the time, constantly purring and climbing on our laps and today he actually played a little (so far he hasn't been playful at all) it's just those few minutes of the day he seems absolutely desperate to go out. I just needed some reassurance that I wasn't being awful keeping him inside even when he's crying to go out. I'm hopeful he will adjust relatively well as he is so affectionate and loving and is relishing all the fuss he is getting.

Strays who once lived indoors and were put outside and abandoned usually adjust fairly easily to being back inside. If you let him back outside, you run the risk of him getting hit by a car or getting injured or run off by another cat. Of course there are many people who do allow their cats to come and go outside, and depending on where they live, this can be a workable situation, but I wouldn't even consider doing this until after he's neutered and vaccinated.  It sounds like you are doing a fantastic job with this little guy