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?
From TFD
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.
Best Answer
Use her name - in the UK, there is no expression you could possibly use that would not be considered insultingly paternalistic (if used by a man), over familiar, patronising or chauvinistic. In previous times, terms such as 'dear' 'love' 'duckie' 'sweetie' 'sugar' and even darling might have been used - none would be considered acceptable today in the circumstances you describe.