Just read this heady but interesting essay on the question above: http://www.firstpersonscholar.com/identification-or-desire/
Should be read in its entirety, but it does wrap its point around difficult terminology, so I'll summarize best I can.
The point is essentially that while traditionally academics, and many players, have considered playing as characters in games to be about identification, seeing ourselves in those characters, or non-identification, completely separating ourselves from characters unlike us, this may not be accurate.
Instead, Voorhees argues that both people who identify as men playing as women avatars and people who identify as women playing as man avatars are enacting a queer relationship. A man playing as a woman is enacting a transgender-like experience, placing himself into the context of a woman's perspective.
Of course, being transgender is not a playground or a choice, but a realization, or at least actualization, of the gender you feel yourself to be, a gender that you were not assumed to be by society up to that point. So it's not the same.
Voorhees and the discussant at the bottom do go into how if it is true that playing as someone different creates a queer embodiment, then that may result in people uncomfortable with that idea pushing back with less exploration or more vitriol, something the discussant compares to men acting more heteronormative when in homosocial situations, like everyone being unclothed in a locker room.
The point of all of this is is: A study referenced in the article shows most males playing as female avatars do so to gaze and control them, a domineering relationship. Do you, in general, feel that you embody the character you are playing, becoming a different or similar person to yourself, be that in race, gender, sexuality, ethics, or personality? Or do you only identify with similar characters to you? Or do you choose a character to gawk at? Or is it a more complicated mix?
Personally, I try to create a cooler version of myself in sports games since it's living a fantasy. In role playing games, I want to play as something other. I try to embody, for instance, a female avatar as if I were her. I find it frustrating when something like Skyrim does not so much let me be a woman or a man, but removes most gendered existence from the world, making the character an it with a gendered pronoun. More active commentary and use of both positive and negative gendered experiences may be something essays like this one can help us encourage in those who write games. Portal 2 is very gendered. It uses some seriously negative gender tropes in Glados' insults, but it does have some major moments that would not work with a male Chell.
Thanks for reading and possibly responding.