Wow. I read the question about ten hours ago and it has been on my mind since. It's a great question, and after a lot of thought, I think I have my answer.
I'd go with Wally Lamb's "I Know This Much Is True", which I just read last summer. It's a long book, over 900 pages, and I was not thrilled with the way it ended, but I identifed so much with Dominic, the narrator and main character, that it really affected me.
The thing is, I can't tell you why I identify so much with Dominic. We have nothing in common. Dominic is a twin (his brother is mentally ill), divorced, a father grieving his dead infant daughter, trying to figure out who his father was, and coming to terms with his relationship with his borderline abusive stepfather. And that's just the start.
Needless to say, my life is nothing like that. But as I read this book, I felt that the author knew me every time I read Dominic's words and thoughts. We share a lot of the same opinions on society and are always trying to figure out why things have happened and will happen.
There is an overlying theme in this book about self-reflection; people who would benefit from it but can't bring themselves to do it, people who ignore it, and people who benefit greatly from it. Dominic learns a lot about himself in this book, understands how some of the things that have happened in his past affect his present, and why he has the relationships that he does. I feel like I have spent the last few years of my life in a state of constant self-analysis, and there were many times where I felt an eerie connection with Dominic.
The book I WISH described me to a stranger would be either Hemingway's "The Sun Also Rises" (Jake-minus the, ahem, "war wound") or "A Moveable Feast" (Who doesn't want to be a part of a lost generation?), but that is pure fantasy.
Extra credit: the book I did not like that most people said was great? Easy-"The Bridges of Madison County." Maybe not because it's a bad book (I think it's average) but more so because the way people described it to me, like reading it would mean that the skies would open up and I'd immediately ascend into Heaven. I kept waiting for the "revelation." I'm still waiting.
And I ruined the movie for a few people when I saw it in the theater, but that's a story for another time.