There is a lot of discussion about this in Pacific countries where there is a social construction about feminine males and masculine females. It is noted in these countries there is less room for accpeting same-sex relationships as western countries.
From my perspecitive I think due to the sexual dimorphism in humans is what leads to gender. Males are significantly bigger/stronger than females which means that in terms of civilisation males are able to complete more physical labour than females of the same age. When this became linked to surivial that is where "gender" arose from.
Because humans have significant mental capacity where we can conceptualise individual nervous output feelings rather than react to behaviour, it would also make sense that for some people their feelings around social roles would be outside of their bodies. As some of that conceptualisation has been spiritulised then I can see why some people would consider this a soul and that soul to be strongly linked with their self.
Where I think gender ideology gets it wrong is they have taken the Descates "I think therefore I am" to an unachievable extreme. They have have created a ideologic structure permitting people to have believes that their gender is the reality, rather than their body. What they haven't done is put the guardrails up of this is my reality but it does not reflect reality for everyone else.
Most people can do this themselves, we are a social species and therefore it benefits our survial to get along. I have no doubt that most Transwomen especially are horrified at the - misogny, same-sex sports, forcing women to share spaces where they are at risk, and the transitioning children- being conducted in their name.
The people that cannot see outside of their own feelings are the people predominately on social media and in politics. There was research on narcissistic tendancies and their said that actors have a high level of narcissitc traits but that gets kicked out of them in the training and auditioning process. Influencers it apparently gets worse, making it look like the majority of trans people.
TL:DR I don't believe in a gendered soul but I can see why some people do. The issue isn't what people feel, but how they expect others to see them.