A baby is assigned a sex/gender at birth, but when the baby grows up it may feel a different gender from the one it was assigned at birth. But many of you appear to think a person cannot change gender, with all the references to a 'man in a dress and lipstick' etc. It isn't just about the biological sex we are born with.
You are confusing the words sex and gender. You cannot debate this unless you understand the difference.
Sex is real and immutable. Sex in humans is established at the point of conception. Before the woman even knows she is pregnant. It is not 'assigned' by medical staff. When the child is born, it's sex is observed and recorded. This will always be the sex of that human. It cannot be changed, even by medical intervention or hormones.
Gender is a social construct. It is different depending on where you live in the world. It's based on how other people see you. You can change your gender by signalling to others that you wish to be regarded as a different sex. A 'man in a dress and lipstick' is signalling in this way. We can still see that he is a man but we can also see that he wishes to be regarded as not a man. This is transgenderism and because it relies totally on affirmation from strangers, it's a very difficult way to live. Some people, like Eddie Izzard for example, 'change' their gender by changing their clothes and therefore changing their signals. Eddie calls this boy mode/girl mode.
Many women do not like gender stereotypes being allocated to them because they are regressive and harmful (think of 1950s woman, womans place is in the home, unequal pay, weaker sex etc.) Because of this, they are critical of gender stereotypes (GC). This means that they don't believe being a woman is all about hair, clothes and makeup. And they certainly don't believe changing clothes is literally changing sex.