It’s classified as a sport (a competition between two people or two groups of people). In this sense, Chess is also a sport.
Sumo wrestling is a sport except sumo wrestlers are at least twice the size of a darts player. Besides which there are thin and medium sized darts players. Fishing is classed as a sport but I cannot see the point of sitting on a river bank for hours on end with the company of a plastic tub full of maggots.

Darts is not a very good sport for keeping yourself physically fit and some of the professional players are overweight, but this however does not stop the game from being classified a sport.