I'll take identity as meaning two things here. Physical identity and emotional identity. physical meaning i can create a character that looks, sounds and smells exactly like i want him, ranging from hair colour and left nostril width.
Emotional meaning being able to make the decision that I, the player, would make in the stead of the protagonist.
1) depends on the type of game. A game like WoW or Skyrim for instance, thrive on letting players be who they want to be, because their story revolves around a hitherto unknown hero rising to meet the enemy. This is inviting you to play as you, or an avatar of you.
A first person shooter places less importance on the protagonist being you, but instead try to make you empathise with their character and the decisions they make storywise. Prime example that comes to mind is Deus Ex, where there is nothing you can change about Jensen except the preferred playstyle and skills, but they try to make you feel what is going on with his personality.
third type would be a final fantasy type game, where apart from the name, most of the story is written out for each character and you are taken for the ride. Some characters would be written to be liked and some to be disliked, but their role is already mapped out and has nothing to do with how you feel about it.
So yes, identity is free, if you pick the game that you can be free in. I like making a character how i like him to be, but if a character is well written, i can also just enjoy his story. Taking into account that i am a white, male 20 something, so most protagonists are already made to look like me.
2) No. While race and gender is usually well represented in skyrim for example, it doesn't leave a lot to mess around with. What if i'm a person born as a redguard, but born and raised in the imperial city, with it's stats and perks? What if i'm the only human ever to become a shaman? and this is just talking about race and gender. Sexual orientation leaves a lot more to be said. Even between good and evil there is no middle, grey area that most of us would walk. Making a small decision that benefits me hugely when i need it, but is considered eveil, turns the game ending around to make me look like a terrorist that eats kittens and sets fire to orphans. I know of no game that judges my actions fairly, but i might just not have played it yet.
3) The entirity of the games industry doesn't have to do a thing. it is coming along nicely. The creators of the Infamous games for example are experimenting away with morality, and have shown developers that this is a thing now. Just like Mass effect, dragon age and Knights of the old republic have shown us that sexual orientation is a thing that does not have to conform now too. I would like to see more characters being genuine, but it's coming, i'm sure of it.
BTW, Norr, are we writing your PHD or something?