Imo:
Sex is biological fact - XX/XY. You can't change it.
Gender = social construct, performative behaviour. Stereotypically-feminine behaviour is different depending on your culture, your age to some extent, your social class.
Gender is a binary spectrum, with extreme masculinity on one end and extreme femininity on the other. In the crudest terms, for explanatory purposes: Masculinity in my culture is associated with physical strength, taciturnity (usually), dominance, hairiness, muscularity, plain dark boring clothes, authority, short hair, no makeup. Femininity is about the opposite - submissiveness, softly spoken, looks pretty, long hair, makeup.
I don't know if anyone has a really fixed gender identity. Maybe they do. Personally, I probably move up and down the spectrum throughout the day, each day. Fighting in a martial arts class is not a feminine behaviour. Neither is scratching my arse while watching the telly. My husband listening closely to me banging on about my feelings and making empathetic comments is not particularly masculine of him. This is because femininity and masculinity are silly stereotypes and not at all an essential part of our biology or born in to us at all. Sometimes gender stereotypes actively conflict with the "corresponding" biology - giving birth, for example, is one of the most intensely female things a person can do, but it's not at all feminine.
Tldr: gender is all bollocks.
OP - I think your daughter is correct. She can change her gender any time by shaving her head, wearing clumpy boots and openly farting. That's how shallow and meaningless gender is. But she will always be female.