I don't understand how a person ceases to be a male or female just because they have the wrong sex organs. There is more to gender than sex organs right? I'd be more inclined to look at asexual people as a screwed up version of a male or female, not a distinct gender.
Yes, gender is definitely more than sex organs. It is a complicated set of behaviors and expectationds forced on people by the societies they live in.
If a person is born without any sex organs at all, then they are asexual by default, because people cannot force them to be male or female in gender by expecting them to behave in a "male" or "female" way. So there is some physical basis for gender, however it becomes more than that because of society's expectations.
If someone wanted to discover who they were as a full person... it might be an interesting experiment to figure out how much of their self-identity is caused by societal perceptions.
It is also definitely not a choice to be homosexual, btw.