SOLIDERS possess super-human equivalent abilities. There can be some relation to the Aeon in terms of specialties that can influence and alter the SOLIDERS personality and some such later on. Right now, we're in the preliminary stages, "Tier One" of our bad-assery.
As for magic and special attacks... Let me think on this, because you propose a valid point. I'm thinking it'll be applied as a low-level type, again "Tier One", into your skill sets. It can be linked to the Aeon, like Mr. Hobo's scythe deal.
They're unaware because of their minds being swiped and purged about the experiments and alterations that went on in Fort Lullin. The SOLIDERS think it was just some form of special training they endured before coming back to the capital. Some might feel strange, or that nagging itch in their minds, but who knows what happened, really.
Our Limits/Overdrives are only seen in battle, and some might be unaware they even have it. Others might receive a peculiar twinge that tells them to do something and suddenly there's giant, flaming sword descending from somewhere.
It's more like a weakened, projected form of the Aeon, but I get what you'r referring to and I've already discussed that with the player. They'll amend that. c;