The assignment of sex at birth refers to the determination of the that a newborn's body appears to reflect. That assignment becomes the basis of expectations regarding the child's medical needs (especially beginning at adolescence) as well as the assumed gender identity (assigned birth gender) of the child.
When sex is surgically reassigned, the genitalia are altered in ways which remove some or all of the functioning of the birth sex and take on some of the functionality of the opposite sex. (With further medical progress, it will likely one day be possible to provide all functionality, including reproduction.) This effectively alters the sexual functioning from that of the patient's sex at birth to that of their preferred sex.
There may also be some confusion between "assigned birth sex" and "assigned birth gender". The former refers to a determination of what is present, while the latter is a best guess based on the former, since there is as yet no objective way to determine gender identity.
It's also important to note that sometimes even the initial determination of sex is wrong or debateable. Two obvious causes of this are:
- complete androgen insensitivity syndrome, in which a baby with XY chromosomes appears female (and typically identifies as such), with infertile testes
- 5α-Reductase deficiency in which a male baby appears female due to unusually small male genitalia, which grow to a more normal size at puberty which then progresses normally for a male
These and other exceptions to the assumptions inherent in sex assignment are often dismissed as "edge cases" – but if they are edge-cases, then so are transgender people.