This is a highly personal choice that he gets to make. Some service dogs understand the difference between working and playing. Some don't, so you have to be more strict. His dog wouldn't have made it through training if it didn't want to do the work. I had a buddy whose guide dog was on leash literally all the time. If he wasn't, he would just stand there and bark at you. I would say good on your boyfriend for making a choice for himself to give himself the type of working team that he wants. Too many people let their guide dogs slip in training and become pets. It sounds like your boyfriend doesn't want to be one of those people.

Considering how new the relationship is, I understand why he's being a little on the stricter side. Its far easier to introduce new bad behaviors to a dog than it is to attempt to get the dog to unlearn those same behaviors. Consistency is key, particularly when a dog is young, and the working relationship is in its first year.
Apart from that, I can't really speak to the needs of this dog, or its personality. Some dogs function better with extreme structure. others do best with a little more freedom.
Its highly unlikely the dog is truly unhappy though. Guides select themselves for this kind of work, and far more fail out of any program than make it to the end.

It is still early in their relationship, so that might be playing into his way of working with his animal. This could change as he and the dog grow together and the dog matures. Don't worry about the dog, these guys get satisfaction from working, and they have been conditioned differently from other dogs. I let my dog off the leash and out of his harness at home all the time, and 75% of the time he still sits or lies at my side and follows me throughout the home, even when he is playing with his toys.

My SO has a service dog and she tries to be pretty controlling of him when they’re out, but at home she lets him off the leash and harness and just lets him roam around and play. I don’t think it’s necessarily wrong to let them off their leash and have fun when they’re not working, but she’s also had him for about 8 months and he was pretty well trained before then.