OP wants to "explore the reasons why" some of us say we don't have a gender identity. Well ...
I will have a go at explaining.
I don't have a gender identity in exactly the same way I don't have a guardian angel, a paranormal aura or a method of squaring the circle (with straight-edge and compass ...) ... not in the way I don't have blonde hair, a million pounds in the bank or a friend in the Egyptian government.
What's the difference? Well, in a way I could have had blonde hair; in a (different) way I could have a million pounds or a friend in the Egyptian government. But I could not have had a paranormal aura, a guardian angel, a method of squaring the circle or a gender identity. Why? - Because there are no such things as guardian angels, paranormal auras, methods of squaring the circle, or gender identities.
Someone might delude themselves and think they have a friend in the Egyptian government (etc). (Perhaps they have an Egyptian friend who tells convincing lies. Whatever.) That is a different kind of delusion from thinking they have squared the circle, or that they have a guardian angel or a gender identity.
If someone tells you she has a gender identity, you know she is lying or mistaken, just as if someone tells you she has a method of squaring the circle.
Again, for emphasis: there is no such thing as gender identity. No matter what some people may claim, no matter how many people believe the contrary, there is no such thing.
So, to repeat, no, I do not have a gender identity. It would be impossible for me to have a gender identity because there is no such thing.
I hope that is some help, OP.