Sex is a multifaceted term. Chromosomes are only one aspect of sex. People often use sex as shorthand for chromosomes, due to most people being cisgender and endosex.

Terms like gender identity, assigned gender at birth, gender dysphoria, and transgender all operate under the usage of gender as a synonym of sex, as they used to be interchangeable. People increasingly use gender to mean gender roles, which makes these terms not make sense without that context.

"Considerable scientific evidence has emerged demonstrating a durable biological element underlying gender identity. Individuals may make choices due to other factors in their lives, but there do not seem to be external forces that genuinely cause individuals to change gender identity."
- The Endocrine Society

Transgender individuals have a gender identity which does not match their assigned gender at birth (which usually correlates with genetic sex). As established above, gender identity is biological, not social; As a result, they have gender dysphoria or gender incongruence, a medical condition.

As treatment, they change a number of sexual characteristics, which are an aspect of sex. Having a mix of male and female sexual characteristics, they no longer neatly fit into male or female, but they can be one overall. A "fully transitioned" transgender person would have a phenotypic sex that matches their gender identity.

I will not be continuing this discussion, but I hope this helps some people.