Glad you had him tested for diabetes and are aware of the link between constant thirst and drinking and diabetes. Keep an eye on that just in case because diabetes is nothing to mess with, and the doctor should be willing to retest any time you want. Was it a "fasting" test or just done whenever?
Does he seem to be truly thirsty when he's always asking for more water? If so, that would be more of a sign he might be diabetic but your description indicates maybe he is drinking for comfort and out of habit, since his cup is his lovey and is always in his hand.
I would start reducing the water. He's going to ask for more. Just smile and say lightly, "Oh, sorry, no more right now" --and then be ready to distract him immediately: "Let's go outside" or "It's time to leave for X" or whatever. Have things he can do with his hands, besides drinking from his cup, ready in every room and in the car as well; when he asks for more water, instead hand him something tactile like an age-appropriate play clay or coloring set. He needs to be distracted consistently to get him used to doing something that makes him put down the cup.
Also, was he using a pacifier prior to going so crazy over his cup? It might be that he likes the sucking sensation of the cup, if it has a mouthpiece he has to suck on to draw out water. Even if he doesn't technically have to suck on the cup to get water out, he might be sucking the top anyway and possibly chewing on it too as comfort or for teething -- not great for his developing teeth. But it might explain the constantly having a cup at his mouth.
I would focus on this for a while and be sure he has things to do (and that you have time to be next to him doing them) so that he gets distracted from the cup. You might even have to "lose" it before an outing or two, so he is going away from home to do something fun but the cup isn't there -- eventually he's going to come home and the first thing he does will NOT be to ask for the cup.
Do you go places regularly like a church or house of worship, or the grocery store, or a play group, etc? Does he currently carry the cup and drink in all those places? If so, I would now make some places drink-free zones. (This is a good idea anyway since he's old enough to start learning there are places we drink and eat, and places we do not.) Maybe start with one place, such as no more drinking during church, then build up to leaving the cup in the car at the store. Maybe you already do this?
I would note that the person posting below about too much water intake is right. Hard to believe but yes, it is possible to take in too much water -- actually happened with an adult friend who was trying so hard to be hydrated that he overdid it and the doctor said he'd thrown off his body's chemistry. (He was fine.) Not saying that's your son's case but just be aware of it. Most of us, including most kids, do not drink nearly enough!
I do think it sounds like drinking has become your son's comfort, like a pacifier or blanket for other kids, but it's a comfort that has a lot of potential issues, and once he's a bit older and maybe in preschool (is he now? Does he drink there or does he get distracted by the activities?) he will not be allowed to carry his cup around.