Since a Dread would be a creature within 10 feet of itself, would it be affected by its own aura?
The short answer is that NO, it does not cancel out their own immunity. As people have pointed out, this is a problem beacause "aura" has not been defined in the rules.
We will add errata to our powers and abilities to denote this, so that things become clear.
Aura of Fear states:
As the character is not her own enemy, then no, this does not affect her.
An Aside about RAI:
I suppose you could read the following sentence in isolation of context and come up with the idea that it does:
However, there's precedent for the idea that negative effects don't usually hit the person who cast them even if they're in the center of the effect. Most other "Aura" abilities specify; for example, Elemental Aura:
(you cannot be adjacent to yourself)
(you are not hostile to yourself)
Couple that with the preceding sentence which specifically calls out enemies as being effected by the aura and you have what I believe to be the intent: the aura only effects enemies, period. You and your allies do not lose immunity to fear.
Dreamscarred Press Specifics
It was pointed out in the comments that my examples are core Pathfinder material from Paizo while this class was written by Dreamscarred Press. So I combed the Psionics material and found more examples of this kind of confusion. Corrosive Aura:
By definition you always end your turn in your square. So you'd take 1d6 damage a round.
Worse, Regenerative Aura clearly intends to include you:
Untouchable Aura is the only one written to exclude you:
(Unless you attack yourself).
If I were the DM, I'd just rule that DSP is terrible at explaining what they mean and go with the solution I offered earlier, because this stuff is giving me a headache, but there's evidence to support a ruling the opposite way on both this and Corrosive Aura. I'd not rule one of them one way and the other the other way, I'd go both or none.