I hadn't really thought about this, but I know I had some sort of awareness. These days, I only buy games which I can create a custom character (or simulators where there's no visible protagonist). Admittedly, it's pretty rare for me to play a female character in said RPGs. I'm a role-player and a man, and as such I have a better instinctual sense of how to role-play a man than a woman.
However, on one playthrough of Dragon Age: Origins, I played as a human warrior who was a woman and got romantically involved with Alistair. Admittedly, it was a little strange at first, but once I got into the role, I felt a sense of attachment to him that comes with a relationship. He was an anchor in the all-over-the-place life the protagonist lives and was a source of comfort and safety as such. It wasn't all that different than how I felt about his being my best friend when I played as a male character.
However, on one playthrough of Dragon Age: Origins, I played as a human warrior who was a woman and got romantically involved with Alistair. Admittedly, it was a little strange at first, but once I got into the role, I felt a sense of attachment to him that comes with a relationship. He was an anchor in the all-over-the-place life the protagonist lives and was a source of comfort and safety as such. It wasn't all that different than how I felt about his being my best friend when I played as a male character.