Hrmm ... yes, but it depends on the game.
Walking Dead, yes. I'm a nutjob zombie fanatic / closet-survivalist, I try to consider myself a 'decent' guy, I have a background in education (not in education anymore), and I have a tendency to assume leadership roles - and I also have kids not too far in age from Clem. In other words, I viewed the game as a trial run for the impending zombie apocalypse and made decisions exactly as I would given the same situation. I love the addition of the "let's see how everyone else chose..." at the end of the chapter. My purpose became keeping as many people alive as possible, and if that meant leaving 1 behind it became an easier and easier decision to make.
Skyrim, no. In fact, in most 'fantastical' RPGs I make a character at the start with a specific character in mind: I've got my 'noble savage' Nord 'Conan' type, my vicious Imperial enforcer a la The Hound, my heroic 'all around good guy' Redguard (akin to Cyrus), and my roaming alchemist/summoner Breton. But a key factor is that the RP elements of Skyrim are rather poor - interaction with NPCs was basically 'Press X to hear the next thing they have to say'. Hell, in the main Alduin storyline there's only really three points you can actually make a decision that affects the quests - one, follow the Nord or the Imperial in Helgen, two, go to the Blades or the Greybeards, and three, kill Mr Talkative Lizard or don't. I do however play as myself in Fallout - I try to make decisions that I would make (again, as a survivalist nutter) and I try to deal with the factions and missions as I would myself. But I've been playing the Fallout games since #1 and I've always played it that way.
EDIT:
Ooh ooh ! I forgot the greatest of them all!! Vampire:The Masquerade:Bloodlines. Now that I *ahem* 'insert myself' into. Especially the bits with Jeanette. I insert myself as many times as I can.