Disclaimer: Not an answer, just what I managed to come up with.
Try as I might, I can't solve this one, so here are my notes if it helps somebody else solve it. I'll update this if I figure more things out, but right now I'm stumped. I feel like I either don't have enough information or just don't know enough to solve it. (It's a fantastic riddle though! I hope somebody can figure it out.)
My working theory at this point is that the answer is a number.
I found myself to be the fifth, although I was more part of one
The answer is obviously going to be the fifth of something, implying that it exists in a sequence of related things. I think it means the fifth number in a sequence of numbers, maybe the Fibonacci sequence (1, 1, 2, 3, 5...) or the prime number sequence (2, 3, 5, 7, 11...)
It all started with a break of peace and a Greek gift
The second part of the phrase, "a Greek gift," is probably a reference to the Greek gift of the Trojan Horse, and the famous saying that comes from it, "beware a Greek gift." It typically means a curse disguised as a blessing. But I'm not sure how this relates to anything, or what the "curse" could be.
I was born, source unknown, with powers raised
"Powers raised" immediately brings exponents to mind, since numbers with exponents are typically described as being "raised to a power." Coupled with the first line and the other mathematical hints throughout the riddle, I think this means that the answer is a number, and that exponents/powers are involved or important in some way.
It took a scarf to know the story’s whole
No idea what scarf this could refer to. I've been trying to find references to scarves in mythology, etc. but nothing seems to meld right. But if my number theory holds water, then maybe "whole" could be a reference to whole numbers (integers)?
I fought three valiantly, and hid from the fourth
Obviously another numerical clue, and because the wording of three is vague, I feel like it could refer to either:
the third and fourth items in the sequence of objects that the answer
is the fifth of, or
the first three objects together and then the fourth object.
Don't know what it means beyond that. Perhaps we could get some hints?