It is unique because FIDE is the only federation to define chess as a sport and not as a game.
More seriously, chess looks much like checkers or go in what is always indicated as its 'uniqueness' : same starting position but always a different game, no chance involved in the battle of two minds so you are the only one to blame if you lose, easy to learn and hard to play, etc.
Why is chess unique from other mental games like the Rubik's cube and sudoku puzzles? What unique experiences does it give you that no other mental game can offer?