Are you mouth-breathing a lot? That could increase dryness in the mouth and promote thirst.

Also (not to be nosy but...) how's your urine output? I have a poor sense of thirst so I judge my need for water intake solely on frequency and color of urine. It should be a very pale yellow tending to clear color, but not more frequent than 60-90 mins. If you're going once an hour or even more often, you may be drinking too much and you could become electrolyte deficient. However if you are drinking that much and not going excessively then perhaps you really do need the water.

I do not run so I can't compare running vs. cycling. However, what I have observed is that cycling has this mysterious way of requiring a serious amount of calories and water every hour to stay properly fueled, whereas I can hike for 3-4 hours just sipping water and then stop for a good snack. Even starting on a full tummy, if I ride for 3-4 hours without eating I will bonk, hard! And I don't feel that I'm working so much harder on a bike than I am carrying a backpack up a mountain, if anything cycling seems easier. Strange stuff.