That certainly doesn't look like it meets any code to me but I'm not an expert. There's a reason the water heater has a vent i.e. to get the exhaust gases out of your home, completely out of your home.

Thats pretty much asap priority before you do anything else. Its like a gas furnace etc. They must be vented.

The vent pipe needs to be connected to the roof cap. It should not be too difficult to replace the vent and connect it properly. The question is why is it disconnected? My guess is the roof deck and shingles were replaced but the roofer did not bother reconnecting the vent pipe.

Mike, there's a patched spot in the ceiling where the previous pipe lined up with the roof vent. It looks like they replaced the water heater with a different sized heater that sits further away from the wall and instead of connecting it they just vented it straight up into the attic.

Would it be a simple chop of the last few inches and attaching a flex section?

Emergency situation. Carbon monoxide poisoning and a fire is no joke.

I agree that you should cut it back to have room to reorient. But do not use flexible vent. Use an adjustable, smooth wall vent pipe like this:

I don't think that is a patch. It looks like black mold. What is happening is in the winter the warm air is condensing against the cold roof deck and creating the mold. Fortunately it is only in the that one spot so it suggest the attic has good ventilation. Even if the hot water heat was twice the size it would not have moved that much. It is possible during the renovation the hot water heater was moved to make more space for something else.

It is hard to tell from the photo, but that does not look like a B vent pipe. The B vent is has a small air gap around to perimeter to the exterior cool. The materials are not expensive. Here is piping available at Home Depot. I suggest you have this done properly and not have to worry about it for as long as you live in this house.

Hire someone who knows what they are doing to replace the whole thing. Do you know the proper dimensions needed so you don't get backdraft? My sister's house was done improperly (I think too large diameter)and they kept getting backdraft of carbon monoxide (from gas water heater) until the proper exhaust installed. Carbon monoxide alarm kept going off in the middle of the night. Her husband kept "fixing" it. My sister called fire department one night and got red tagged and couldn't use water heater until it was fixed and passed inspection. Told her husband he was going to get them all killed with his "fixes".

Mike: Not possible to definitively state, but I think that is (and should be) B-vent. OP - get a heating company in to fix it - mold/carbon monoxide is not to be messed with.