All other forces have some kind of charge system, electromagnetism has positive and negative, weak force has hypercharges, and strong force has colors. Why doesn't gravity have anything like these?
It does! The 'charge' of gravity is 'mass'. One aspect of the equivalence principle says that the gravitational-charge "mass" is the same as the inertial "mass". Why that seems to be the case is unclear. It's also unclear why there is no (or why we haven't observed any) 'negatively'-charged mass, but there is no known theoretical reason why this must be the case.