Cole is like the Shades.
The Shades are demons that tried to enter the world, didn't possess anything, and just "materialized" into the world from the shadows. Unlike Ash Wraiths who... what? Possess ashes?
Cole was likely a spirit, maybe a spirit of mercy or compassion, who got lost in the world, like a demon can, and just became a part of it, like the spirits/demons who possessed trees and such.
So, his original form/being is probably unknown to him, and he is now just Cole.
It may be easy to categorize spirits/demons who are in the Fade as their titles, but things tend to get all muddled when they enter Thedas. Justice can become Vengeance, Lady of the Forest, Cole, that one poet tree, and who knows, maybe the original elves were spirits that took on physical form when the Veil was formed.
So, Cole is just a unique spiritual being who wanders around, trying to find a long lasting motivation to go along with his immortality. The spirits of the Fade find their motivations with the emotions/virtues/vices of mortals, the demons in Thedas find purpose in chaos and control over others, and some that just become part of the world just try to be and experience the world.
Edit: It's not like he is trapped here, he can go to the Fade by being killed, although he would likely have to "gather himself back together" to have an identity again in the Fade, as spirits seem to disperse when killed in the Fade or Thedas. Y'know, go from Wisp back to Spirit, although whether he would still remember anything or even be the same person would be questionable.