My read on it (assuming Quizzy isn't an abomination or something) is the whole argument about Cole is just challenging his ideas about change. He glances at Quizzy and back to Varric maybe because that's what he/she represents? Even in a non romanced sense. I think Solas and the Quizzy are implied to have some sort of connection (unless Quizzy just has awkward shared dreams with everyone
). Whether as friends or a possible villian relationship..or just someone who challenges his view. I really don't think he expects to make friends with the Inquisition. Remember he's there to recover the orb, etc...
If the orb was indeed Fen'Harel's (theough there are some indication that it might have actually been Mythal's) then there's obviously a connection, especially now, considering that now Quizzy is the only thing left after its destruction. I think however that it's about something else. I mean, sure, Solas didn't expect to find anyone who he'd befriend or greatly respect OR challenge his view on Thedosians, but still that doesn't really fit the context of the scene and what they were discussion. They were talking about changing someone's very nature, not someone's views.
As for the "she's real" thing. I wonder if its just an inside joke? Cole and Solas break the 4th wall a lot in their banter. Or maybe Bull was right about him and spirits, and he doesn't have too many "real" friends? He's too obsessed with the fade? 
I don't think it's the inside joke, considering that it's from the banter after Solas ended relationship with Lavellan.
Cole: (If the Inquisitor kept the vallaslin) Stop. You are perfect exactly as you are. But then you turned away. Why?
Solas: I had no choice.
Cole: She feels her face, marked, marred without malice. She didn't know. She thinks it's why you walked away.
(Or)
Cole: (If the Inquisitor had the vallaslin removed) Ar lasa mala revas. You are so beautiful. But then you turned away. Why?
Solas: I had no choice.
Cole: She is bare-faced, embarrassed, and she doesn't know. She thinks it's because of her.
Solas: You cannot heal this, Cole. Please, let it go.
(and if you pick:)
Inquisitor: Perhaps Cole can get a better answer from you than I did.
Cole: He hurts, an old pain from before, when everything sang the same.
Cole: You're real, and it means everyone could be real. It changes everything, but it can't.
Cole: They sleep, masked in a mirror, hiding, hurting, and to wake them... (gasps) Where did it go?
Solas: I apologize, Cole. That is not a pain you can heal.
Cole also has many dialogue lines when he says that "he's made himself real" and the like and I think he rather clearly means "real being, able to exist in real world".
He disapproves when the Quizzy says they are like anyone else because the Quizzy is unique. He warns the Quizzy about false humility and giving up your power carelessly later in the convo after the well.
Yet in the balcony scene, at least with Trevelyan, when we respond "Thank you. I try." Solas commends their modesty.
Besides - if you say "I don't think I'm different from anyone" he doesn't mention anything about modesty, he only says "Perhaps in the form of your body, no."
Like I said, seems that Solas refers to either the strength of unique qualities of Inquisitor's spirit (soul?). Though in what context exactly, it's hard to guess at this point.
I remember musing some time ago in one of the wilder speculation threads that maybe there was a time in Thedas' (pre-Thedas?) history that existed before races - or even gods - emerged, where everyone was beautiful, bright and wholesome, only for changes to the world either decimate them or maybe prevent them from being born/reborn (if we assume that the way sentient, physical beings are 'born' are through the process Cole is going through) or made them small/incomplete/petty. That COULD explain why Solas is so baffled with the existence of Inquisitor - the one soul that apparently slipped through to the world too murky and bleak for it to either exist or thrive. Yet it did, she/he is real, therefore others could be reborn too?