A(n often male) paternal figure could use the term 'son' in a fatherly way without referring to his biological Son.
Imagine a man has been verbally abused by a customer at work. His manger might say to him:
Come here son, I have something to tell you about rude customers
Emphasis on the word I want gender-swapped
Is there a feminine equivalent for a subordinate that you respect?
son (sŭn) n.
1. One's male child.
2. A male descendant.
3. A man considered as if in a relationship of child to parent: a son of the soil.
4. One personified or regarded as a male descendant.
5. Used as a familiar form of address for a young man.
6. Son Christianity The second person of the Trinity.
Is there an equivalent for talking to a woman in a fatherly way?
The equivalent for TFD entry for daughter does not qualify it as 'familiar'
- (often capital) a form of address for a girl or woman
Further more often such 'father-daughter' language patterns are negative, dismissive and lack respect.