The Player is the official 'Warrior of Light'. To be fair that is a label given by the public so really the fact only the Player is recognised as the Warrior of Light is because no one else has been yet. There can be more than one Warrior of Light at a time.

Storywise, the Player seems to be specifically Hydaelyn's chosen champion. Both Midgardsormr and the Ascians seem to suggest as much. She has many agents but the Player is a special case and Hydaelyn invested a lot of energy to protecting the Player while they grew in strength.