A lot of characters that I "identify with" aren't all that similar to me at all, but I can see where they're coming from. Which is rarer than you'd think, because I tend to have a problem with "illogical" characters in movies/books/TV -- not that they're illogical but that thinking that is my first instinct, because I can't relate to them. (This is much more of a problem for me with female characters than male characters, and 99% of the time the problem is with romantic heroines, in movies or books.) So when I really relate to a character's thought processes easily, that's a moment to be savored.
Almost always, the characters I identify with are thinkers -- and usually introverted thinkers. Often they're moody and dark, and usually they're the love interest to the character that I'm SUPPOSED to identify with. e.g. Mr. Rochester from "Jane Eyre".
The only character that I've read/seen and gone "That is ME" was Hermione Granger. Which makes sense, since for the entire series she was roughly my age, similar appearance (not to Emma Watson, but to the description in the book), just as academic and perfectionistic, tough, assertive... and xSTJ.