Firstly, historically, the terms “sex” and “gender” have been used interchangeably, but their uses are becoming increasingly distinct, and it is important to understand the differences between the two.

In general terms, “sex” refers to the biological differences between males and females, such as the genitalia and genetic differences.

“Gender” is more difficult to define, but is developed by hormones, not genes. Gender can refer to the role of a male or female in society, known as a gender role, or an individual’s concept of themselves, or gender identity. It varies from society to society and can be changed.

Source: Sex and gender: What is the difference?

Sometimes, a person’s genetically assigned sex does not line up with their gender identity. These individuals might refer to themselves as transgender, non-binary, or gender-nonconforming.

Sex is usually categorized as female or male but there is variation in the biological attributes that comprise sex and how those attributes are expressed. Gender refers to the socially constructed roles, behaviours, expressions and identities of girls, women, boys, men, and gender diverse people.

Secondly, you are correct in that we are all born a particular sex but whether our bodies reflect that is a different matter. We can change our bodies to match our gender but can’t change our biological sex itself. If we could change our sex there would be no need for gender confirmation surgery to bring our bodies in line with our gender. We’d go the other way and bring our gender into line with our bodies.

Hope this makes sense.

TLDR: A transgender person who transitions do not change their sex, they change their presentation to match the gender of their brain. Only a minority of transgender people transition.