I can't tell. Aggressive playing or attacks?

The younger dog's behavior isn't acceptable behavior, and isn't usual puppy behavior. What breeds are the dogs? You are right to be concerned. Depending on the breed (and I'm guessing this is a large breed dog) he may also be going through a "teenager" phase, testing his boundaries and trying to assert dominance.

Will he sit on command? If not, I can give you pointers on how to teach him. Once he has learned that, have him sit before he gets anything he wants -- go outside, come inside, meals, attention, etc. I also recommend no tug-of-war or other rough play. If possible, you may also want to keep him from coming up on the couch with you. Present yourself as the one in charge. Not as a dictator, but as a benevolent leader. He needs to learn his place in the hierarchy, but not in a confrontational way.

Does he get enough exercise (walks, ball play, etc)? He may also have pent-up energy.

I concur with giving treats and praise when he behaves (even if it's for a short while :-D ). If he won't take treats nicely, you can drop them on the floor for him to eat.

The owners need to decide who will be "top dog" between the two dogs. Without seeing their behavior I can't say which should be encouraged to "take the lead." It sounds like the younger dog is wanting that spot, but the older dog doesn't want to give it up. Usually, it's best to support the older dog in his bid to retain it, but not always.

I would address with the owner. Best case scenario, the dog is spoiled and with some additional work on your part (depends if it's worth it to you) you may be able to redirect the dogs behavior with positive reinforcement. Rewarding with attention or food when behaving as they should. Taking away affection (if you have to leave the room.

Worst case, you or someone else could get injured. Depending on your knowledge/training, it may be an issue you work around---possibly for an additional fee.

Bulldogs are a very stubborn, independent breed. Theres no shame in admitting he's too much dog. Since there is an escalating effect going, I would recommend a different sitter/trainer. No money is worth getting bit over.

The behavior definitely sounds concerning! But without seeing it in person I can't say for sure. It does not sound like normal puppy behavior at all. It sounds like as he's maturing there is some fear aggression surfacing. Lots of things that were okay when he was younger are no longer okay.

I would try to video tape the behavior to show to the owners next time they come to visit. Ask them if they've seen this behavior before, and mention a trainer or behaviorist if they don't seem to know what to do about it. I suggest behaviorist because they are better able to figure out what the dog is going through.

I would agree with Shannon and I have experienced the same breeds together before, although different situation. I would definitely bring it up nicely with the owners. I would emphasize that this dog needs training, and if you're not qualified maybe mention local dog training classes you heard about, biting is not good (so maybe a reactive dog training class). I think this is an important timeframe in any dog's development. Training games with the dog help them to learn boundaries and provide a positive leadership role.