The reason why you don't feel like a Scion of the Seventh Dawn is because you're not. The Warrior of Light, aside from Grand Company affiliation, belongs to no one. However, pretty much every organization in the game swings you around like a weapon to smash whatever threat has appeared at the time.
As a matter of fact, this mutual, but abusive, relationship with the games organizations is actually covered in the Dark Knight quest. You actually learn that your character might possibly be harboring animosity towards the people who are constantly abusing your status as the Warrior of Light.
When your Darkside reveals itself, it's because it has become angered that the people of Eorzea keep pulling you in different directions to solve whatever problems they have at the time. Which, if you think about it, actually shows that your Darkside has more love for you than anyone else. This must be why it's called "Esteem" when you fight it.