I very rarely identify with a character, but when I do, it's scary how similar they seem to me.
The most recent example that comes to mind is when I played through L.A. Noire. Cole Phelps is basically me. Maybe a bit more of my darker and less-logical side than I'd like, but seeing some of the flaws Phelps had actually led me to find those very same faults in myself. For example, he's always very self-assured and anxious to progress-- to a flaw, and he begins to believe the talk that he's a master detective, leading to a larger ego buildup. I'm the same way, I found out. And it was uncomfortable for me to come to terms with this, as I didn't particularly like some of his actions-- actions that I myself would have actually taken were I in his shoes.
I think to truly identify and connect with a character fully, you'd have to at least learn something about yourself through them that you didn't know before.