I think it depends on how you define the word. For much of my life, gender has been used interchangeably with sex to indicate male or female (biological sexes). So to me, gender and gender identity aren't necessarily the same thing. But you can call yourself or identify as whatever you want. Makes no difference to me.

Going by the dictionary definition of gender: "the behavioral, cultural, or psychological traits typically associated with one sex" the implication is that gender refers to male or female or at least the characteristics thereof. You may identify yourself as some other gender but it doesn't necessarily mean someone else will. People will see these traits somewhat differently based on their experience, cultural background and other factors. Most people will see someone else as having predominantly the characteristics of a male OR female or occasionally both. Mostly, these characteristics are defined by nature.

Gender identity beyond male and female still strikes me as made up nonsense or at least poor wording for something that should probably have a different name. But like I said, it makes no difference to me what someone identifies as. Just don't force me into your box.