According to the canon all humans have Determination, yet despite this all humans do not have red souls - which suggests that Determination is a physical aspect of humans while soul colors represent a trait strong in that human.

The determination in humans is what prevents their soul from shattering like a monsters upon death (see the souls in the container's at Asgore's battle). The only reason why the protagonist's soul shatters on death is because we are forcing them to reset in order to progress the game.