I tend to agree that there is something about the Inquisitor that is special, regardless of race. Particularly because Cole's dialog about Quizzie being "bright" comes up no matter what (right? Unless I'm wrong. If so this whole post is trash). I wish I knew if Kieran comments about "choosing that form" for everyone, or if that is elf-exclusive. I would love if someone would compile that scene for all the races and all the options into one video.
Anyway figuring out what it is about the Inquisitor that makes them special... is more difficult. Especially since I can't decide if that stuff Cole says is for flavor or actually going to be relevant. And then is an elven Inquisitor more special, or does the explanation have to apply to all races? If the latter, things become more difficult since Dwarves are not supposed to be connected to the Fade at all and Qunari generally avoid it. Of course they both are due to the mark, but Cole's lines seem to suggest that Quizzie was always the way they are after the mark. Cole's romance banter says "You're real" not "You became real," as if she were always real. So what could be something that makes someone "bright" that can apply to everyone...?
I have absolutely no idea. Could it be something spiritual, like Quizzie is a spirit of X made real like Cole? That seems like too literal a use of the words "spirit" and "real," in the dialogue, plus I'm not sure people would like playing a "fake" character. Were they actually chosen by Andraste or the Maker? That seems unlikely as well, given that BW wants to keep the Maker's nature a secret. Did the lyrium give birth to you? That is just my complete crack theory xD Like I said, no idea. Hope it is elaborated on in the future.
The Anchor probably did something, but I don't think the Inquisitor is a spirit-turned-real. While apparently possible (oh hi Cole), we know the Inquisitor has a family, etc. because we hear about it. He/she didn't just materialize into Thedas out of thin air (though that could have been an interesting plotline).
Solas does use the word "spirit" specifically, and I wonder, as others have mentioned, if the natural state of being is actually to be one part spirit, one part flesh-and-blood, which the sundering of the Fade from reality has caused to be impossible.
Contrary to what others have speculated in-thread about spirits not even existing pre-Fade, Solas paints a picture of them being "as common as grass." It's possible Solas may have once been a spirit, and he can certainly see things others don't. So when he says the Inquisitor has a unique spirit, I'd take him at his word.
As for being chosen of Andraste or the Maker, I think that's up to the player to decide. They will never answer that question directly, because faith is a personal issue. I think the Inquisitor is more of a Chosen of Fen'harel, and that is mostly by accident.
I've been thinking something similar to this, since a Mage Inquisitor can comment during the Fade!Haven scene that the Fade has never looked so "real" in any of their previous visits there, which Solas comments on as being unusual? It makes me wonder if the Anchor somehow allows them to have a foot in both the real world and the Fade at the same time?
It makes me wonder if the Anchor has also made even non-mage Inquisitors into some kind of hedge-mage, as they seem to subconsciously know how to use it on sheer instinct as the magic works through them? And if perhaps they're also somehow bleeding some of the Fade into Thedas with it, possibly allowing them to manipulate reality outside of the Fade, at least on some small scale?
It would explain some of times they survived things that all logic says they shouldn't have, which the Inquisitor seems to chalk up to "divine bad luck".
The anchor almost certainly has something to do with it, perhaps being a catalyst for this change. Or perhaps the Inquisitor was already special (or became such because the player is controlling him/her as madrar suggested?).
The fact that the mark is called "the Anchor" ought to give us some concept of what it is and does, but really it doesn't. An anchor keeps a ship in place--or some other item. So, an interesting perspective might be that the Inquisitor is either being held in place by the Anchor, or that the Inquisitor is actually anchoring the Veil.