Hi there. Welcome to CatChat.
Here's a list of things I'd do anyway- some of which you may already have done!
First off... If he was about 6 months old when you found him, he'd need to be neutered, vaccinated, given flea and worm treatment by the vet (these are the only ones that work- the ones you buy over the counter are not only ineffectual but often quite bad for cats) and thoroughly vet-checked.
He needs to be neutered even if you never plan on letting him outside.
Second... I'd get him vet-checked again anyway, to make sure there's nothing physical causing him distress, or causing him to make lots of noise. People often think their cats are fine because they seem happy but cats are extraordinarily stoical about pain and rarely communicate distress until they're in agony. There are also non-painful reasons why a cat might make noise. For example, deaf cats often make quite a bit of noise because they're not aware they're doing so... And a non-neutered cat would almost certainly be extremely frustrated living indoors and be liable to express this- possibly noisily.
Third... If a thorough vet workover shows there's nothing physically wrong, then we're looking at psychological issues... Is there a pattern to his noise? For example does it start when you go to bed or leave the room? Does it start at certain times of day? Are there other triggers- e.g. something going on outside? (A neighbourhood cat in the garden perhaps). I would try to become very observant about what seems to be starting the miaowing off.
Fourth... Clearly he was living outside when found. And although he's been blessed that you've taken him in and given him a loving home, he may very well be finding the adjustment to indoor-only living is stressful and frustrating. I'm not an expert on indoor-only cats, but there are others here who are... Clearly indoor-only cats need lots of space, interactive playtime, stimulation, and lots of opportunity (and space) for exercise... If you have a garden it's worth considering making it cat-safe- i.e. catproofing it so he can't escape to the road. That way he can go outside safely. There's a wonderful woman on this forum called Roz who makes and sells this and her email is securacat.co.uk... However...
Fifth... You say you're not really supposed to have pets which I assume means that your landlord won't allow it. This worries me greatly because it means that this poor cat could be evicted at any time. If that's the case, then better he finds a permanent home now while he's young and more easily rehomeable. Or else better that you move to accomodation where you are allowed pets. I know this sounds rather harsh given you're trying to do your best for him, but everyone here who works in rescue has seen too many cases of cats suddenly needing to be rehomed because their owner isn't allowed a pet.
Good luck and do keep us updated.
