Some game-designers make games based on the idea that you project your personality onto the protagonist. This can be seen for example in Saints Row 1, where your character never says a word and is utterly without personality. In other games (most games), the protagonist does have a personality, does say things, and I my opinion is much easier to relate to. In games like Mass Effect, you can choose between a few predefined personalities by choosing your characters words and actions. In others again, like GTA4, the personality of the main character is pretty much set in stone.
I have always found it hard to project my own personality onto a character that doesn't have one, and I've definitely found it hard to relate to those characters. However, if you are one of those that can and often do project your own personality onto an in-game character, then I can see how you might find it odd if you are of two different genders. Games (usually) tell you a story through what goes on with the protagonist, and if somebody has a problem with playing a female character that has her own personality, that has no need for you to mix your own into the game, then I really can't see the reasoning behind that. Very few among us will find it hard to watch the movie Alien because the main character is a woman.
A whole other discussion is the one about MMO's. When you choose not to play as your actual gender in an MMO it is no longer just about having a relatable character, because in MMO's there is actual interaction with other human beings through your chosen avatar. One argument is that you send false signals when not playing your gender, giving others the impression that you aren't actually your gender. Another argument is that in an MMO your gender doesn't really matter at all. It certainly has no impact on the game-play, and chances are you'll never meet the other people you are playing with in real life. I am male, and for me to play a female character in an MMO would kind off be cross-dressing. Albeit much more socially acceptable than actual cross-dressing. I personally have no problem with somebody having a character with a different gender than themselves, as long as they are honest about it when asked.