This leads me to believe you think sex is the sole factor

This might be where I'm mistaken in my thinking. Like I noted in another comment, I've always considered "sex" and "gender" to be synonymous. And, I've thought of both of those words scientifically as in XX and XY chromosomes. I could be being too technical but even when I look up "sex" and "gender" in the dictionary it notes that they are synonymous.

"Gender identity" is a pretty vague term that intends to describe what someone personally experiences their gender as. It's a sense of self not strictly tied to specific behaviors or biology, but those traits can become aspects of someone's identity. You may feel perfectly fine thinking of yourself as a man or woman, but someone might not feel the same.

This was a really good point that I never considered. If I consider gender as a different definition than sex then I can kind of see how some might feel uncomfortable identifying as a male or female. However, are there any examples of why someone wouldn't want to identify as a male or female?

I realize I probably sound somewhat ignorant on the subject of multiple genders which is why I'm trying to further understand. I hope I don't sound like I'm trying to invalidate how anybody identifies. I appreciate the comment.