Yes, puppies bite. They don’t have hands, so their mouth is how they explore their world. They CAN learn bite inhibition, but it takes time and patience.
If you’ve ever watched a litter of puppies playing, it typically goes like this. Puppies are playing and having fun - rolling around, running, chasing, and naturally lots and lots of play-biting. As their excitement builds, eventually one of them gets carried away and bites another puppy too hard. That bitten puppy yelps and ends the game. Biting puppy no longer has a playmate, and just learned one tiny lesson about appropriate play and not biting too hard. (It will take many many many more such lessons though, and those MUST continue with the puppy’s human family.) Among dogs, play-biting is normal, they just have to learn what the boundaries are.
With humans, the puppy must learn that any teeth on skin is a no-no, but that is the END goal, not the beginning goal. It’s going to take patience and consistent training to get to that point. You can’t expect perfection, and you must shape the progress. It’s up to you to set the puppy up for success. Don’t let him get into situations where he becomes over-stimulated. If he bites too hard, give a yelp, and walk away and ignore puppy for a few minutes. Then come back and play some more. Praise puppy when he is good. As soon as the biting starts, yelp, and walk away, and so on. Puppies have very short attention spans and short memories. Punishment only confuses them. As long as you’re consistent, eventually the puppy will learn that biting humans means that all fun games stop.
At the same time, it is important to provide the puppy with ample toys that he CAN bite on. This is an important part of his development. Puppies NEED to bite and chew, just not on humans or human property. Don’t just leave toys scattered around though. Give him a toy and supervise what he is doing. You don’t have to always actively play with him, just keep an eye out. Never let him chew/play with toys unsupervised. Also do NOT leave things around (shoes, furniture, etc.) that puppy shouldn’t play with. That means removing valuables from the room or confining the puppy to a specific area. If puppy chews up something when you weren’t looking, that’s YOUR fault, not the puppy’s.
It’s just like when you hear kids playing, and the laughter keeps building and building until BOOM! Now some kid is crying and another kid is getting into trouble. It takes YEARS for kids to learn how to behave. The good news is it won’t take a puppy as long as it does with kids. A puppy’s success in learning how to behave has more to do with the owners than it does with the puppy itself.