I think it's a good question. (It doesn't hurt to ask these things when people spend so much of their lives playing games.) But I think the answer is almost always "no".
The closest I've come to being "myself" in a game is when I've played first person shooters that lacked any sort of character development (other than being able to increase your power by acquiring better armor and better weapons). "Who is this Space Marine I'm supposed to be playing?" Who knows, and who cares? I'm just trying to survive and kill as many monsters as I can and win the game. And do it with style, of course.
My transition to RPGs (Baldur's Gate, Icewind Dale, Torment, etc.) was hampered by my preference for games in which you don't need to "develop" your character and all you need is more firepower. But once I got the hang of roleplaying, the whole point in playing was to create a character in the RPG world, not to be myself the way I am in the real world. Certainly, my in-game choices and preferences reflect who I really am IF I play a character who is like myself. But if I play a character who isn't like me, then it's like writing a story about a fictional character--and I don't think it's true that every character we create is simply an extension of ourselves. If that were true, then fiction would be a lot less...fictional.
Has Jeff Green ever done any sort of literary criticism? Does he have any background in philosophy or psychology? Or is he just a guy who reviews games?