It weighs the least amount on the entire weapons list. Even a hand crossbow is considered a "Light" weapon.
Being light doesn't have anything to do with weight. The definition of a Light weapon is as follows:
Light. A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
Darts are small, obviously. So if a dart isn't a light weapon, it's because it isn't easy to handle, and therefore, not ideal for two-weapon fighting.
Note that while he doesn't say why, Crawford confirmed that darts are not meant to be light. Twice.