Okay, but there are people that identify as someone with a soul, like Christians, Muslims, Buddhists, right? They say they are someone with a soul, and they say the soul is a spiritual thing. If souls do not exist, what are these people doing? How are they identifying as someone with a soul?

Just because someone experiences something internally doesn't mean it objectively exists.

And don't cisgender people identify as their own sex?

You've defined a cisgender person as anyone who has a gender identity that matches their sex, so yes, according to your definition, cisgender people identity as their own sex. The disconnect comes from who then is cisgender and who is not. You are working within a framework where gender identity is objectively real and universal, therefore everyone who is not trans must have a gender identity and that identity must match their sex. It would be like if you were a Christian categorising people into 'saved souls' and 'not saved souls', if you believe souls are objectively true and universal, then if a soul is not in the first category, then it must be in the second. But atheists are telling you there's a third option: the soul not existing in the first place.

I would argue the vast majority of people in the hundreds of thousands of years of human history have not had a gender identity, and only recognised that they were a man or a woman through observing their sexed body. If gender identity was such a universal human experience then surely it would have existed as a concept before the mid twentieth century.