To interpret this as him being mad at you is basically anthropomorphizing too much (that basically means ascribing human motivation to animal behavior).
Animals generally are much more in the moment with their feelings. They don’t hold grudges the way we do as humans.
If your cat were really angry with you there would be no doubt about it. It would be a full-on attack, teeth, claws, the whole nine yards.
The last time I brushed my cat for longer than she was willing to tolerate, that’s what happened to me. It wasn’t just a play bite. She was one seriously pissed off cat ! But once she got free from me and made her point it was over. She didn’t start all over again an hour later or the next day. That’s just not how animals are.
Have you ever seen a litter of kittens either here or in the wild? Once they are old enough to develop their stalking, hunting and killing skills, they practice on each other with play fighting, obviously scaled back a bit.
And even tho our domestic cats have their meals provided for them, they aren’t that different from their cousins in the wild. Play fighting, biting etc. is still part of their instinctual nature.
But in a way, it doesn’t necessarily matter that much why he’s doing the biting. It’s up to you to train him that this is unacceptable behavior.
Hitting him will do absolutely no good and likely make things worse. But you need to make him aware of your STRONG disapproval. Cats are sensitive to tone of voice. They may not understand the words but they can read tone and body language.
You have to decide how you want to get it across to him that this biting won’t get him the positive attention he wants. Whether it’s giving him a time out by shutting him in the bathroom for 15 mins or tossing him off and refusing to interact with him briefly for awhile or whatever else you find works to get the point across.
Pick something and be consistent about it without fail. It’s like disciplining a child. He needs to be able to predict with 100% certainty what the immediate result will be whenever he bites your hand. But the consequence you choose can’t really last all that long since animals are very in the moment.
It may take a while and try your patience but cats are very intelligent and can learn proper manners.
A long time ago I decided that I would not tolerate any cat of mine scratching or biting me and nip it in the bud when they’re young.
A friend of mine had multiple cats and a far greater tolerance for this. Her hand and arms were constantly covered by scratch marks in various stages of healing.
Then one day she ended up spending several days in the hospital getting IV antibiotics for a bad case of cat scratch fever. (This is no joking matter. It can result in a very serious infection. Look it up.)
I decided then and there that I would not be so tolerant of this behavior in any cat of mine. And until my latest kitty, it wasn’t really a problem once they were adults. But each cat is different.
But I have to do my part as well. Some people (guys particularly) think it’s cute when their little kitten play attacks them and start waving their fingers at them to encourage this. When they’re tiny kittens they don’t have the same force or power. When they get older it gets worse suddenly.
I always use toys for my cats to play attack. I never use my hands for anything other than petting and love. And if I go to pet them and they’re in a feisty mood and start swatting my hand playfully, I immediately withdraw my hand and stop everything and wait till they calm down a little. So they get the message consistently that human body parts are definitely NOT for playing with. That’s what toys are for.
Since you found him as a stray you really have no way of knowing what his previous experience with humans has been.
And now that he has been with you long enough to regard you as “his” family, he feels comfortable to play with you as he used to with his littermates or other humans.
My current cat didn’t start playing rough with me (or really playing much at all) right away. It took her quite a while to feel comfortable and secure enough to regard me as family (she had previously been tossed out to fend for herself in the middle of winter snow and passed around to various people before I got her ) She would sleep on my computer desk at night rather than snuggling up in bed with me even tho I encouraged her.
It took about six months before she felt comfortable enough to sleep on the end of my bed rather than the desk.
So, the fact that your little guy feels comfortable enough to play with you like family is actually a good thing. But now you just have to teach him what is acceptable and what is not. If you don’t like how hard he bites then let him know about it and don’t tolerate it at all.
Training is part of being a good pet parent. You’ve done everything else right in terms of vet care, neutering etc. so you’re obviously a responsible caregiver for him.
Now comes the harder part. Training him. Fortunately cats (unlike puppies) come pre-trained for bathroom needs. So now there’s just some behavior modification left and you’ve got a great pet for life.