I'm surprised nobody's mentioned Braid yet. That game certainly gave me a few pangs in the heart.
Lately I've gotten addicted to the TV series Chuck, and have been thinking what it is about the show that engages me so, and how I identify with Chuck always seems to be the most effective element. Yes, I have done computer repair in retail (Circuit City Firedog, we fixed Geek Squad's work

), but that romantic hook between Chuck and Sarah is what keeps me coming back. As spies, they're put in extraordinary circumstances, but that stuff is all just context for the core suspense of the series, which is how they approach all the various combinations of problems that you can have in a relationship and how they work through it. You don't need to be a CIA agent to identify with that.
Braid seems to do this a bit more deliberately, but in a sort of interactive way. Each of the storybooks, and even the strange atomic epilogue seem to capture the pain of lost relationships and growing up. We can't rewind time, but we sure as hell can understand why this guy would want to. The plot isn't even that clear or delivered in a forcibly linear way, but it still manages to hook people.
I think this is why many discussions of story in games always end up talking about JRPGS. Note: I admit that I haven't played many JRPGs outside the typical Final Fantasy titles, Chrono Trigger, and a little bit of Baten Kaitos, but surely I'm not the exception when I say that by the time I got around to playing Final Fantasy 7 & 8 and similar titles I was entering my teen years. These two games are full of all the things that a young male teenager can easily identify with, albeit set in an extraordinary context. People swoon for Tifa from FF7, because she's the childhood friend who grows up to be beautiful and ends up with a bigger, stronger guy than you. Squall from FF8 is a mess of insecurity who seems to have issues in effectively communicating himself in social situations. These are fairly common speedbumps you encounter growing up, and the fact that someone took the time to talk about it makes you feel better about your own particular experiences. Most JRPG developers seem to understand this and create stories to suit their audience.
To get off of the JRPG kick, and to recognize that identification is an element that can be crafted in more ways than one, I want to look at CoD: Modern Warfare 1 & 2. The first game put you in the shoes of a new recruit. You got razzed by your superiors, and found yourself in ridiculous situations such as a sinking ship and a suspenseful sniping mission (yes, different character, but he was in the same situation: still relatively green in comparison to the guy he was working with).
Personally, I'm really not that great at FPS games. I'm usually the guy making dumb mistakes like shooting your friends on Left 4 Dead or stubbornly taking the same route despite getting sniped there every time. CoD 4 seemed to pull me out of the dirt and show me the ropes, culminating in an epic showdown that tested my ability to do everything I'd learned prior in the game. I felt a connection between Soap and myself, we were both sort of thrown into a new environment and grew from it.
Modern Warfare 2, in contrast, seemed to have not only a more difficult campaign than CoD 4, but also seemed to not care about the player as much. Forgive me if I'm missing something (eventually I just got sick of the mission and started playing only multiplayer), but the only mission with remotely interesting plot is the airport massacre, and frankly I can't really identify with being forced to murder hundreds of people to keep a cover. It was an interesting maneuver to play with the fact that you're working in an interactive medium, but in order to get the impact of that moment I feel the campaign's plot was severely lacking in the identification department.
TL;DR: I'll be emotionally moved by any game that's all about me.