idioms - Why 'a friend of mine' is not 'my friend's friend'?

I have some questions about the expression "a friend of mine" and I'm quite confused with it. Actually I have found some threads about this topic but they don't hit my point. I'm not a native English speaker.

General people may interpret that "a friend of mine" is "one of my friends" but it sounds to me like "a friend of my friend" or "my friend's friend" who I may or may not know him/her. I was taught that "mine" is a possessive pronoun and it's used to replace the noun mentioned earlier; for example, This is Adèle's book so the book is hers.

1. Why "a friend of mine" is not "a friend of my friend"? And why "a daughter of mine" is not "my daughter's daughter" or "my grand daughter"?




2. Why we use "a friend of mine" instead of "a friend of me" to mean "my friend" but we use "a part of it" to mean "its part"? "Mine" is a possessive pronoun but "it" is an object pronoun.

I probably have read all things people trying to answer the questions but I still haven't found the comprehensive rules yet. Can anyone give the comprehensive rules for using the double-possessive form?

I myself may conclude that:
1. The double-possessive form is used when the personal subject pronoun is used.
2. If the noun indicating that the owner is a person or people, either the double-possessive form or the noun itself is used but slightly different in interpretation.

3. If the possesser is an animal, robot, or any inanimated objects; the objective pronoun or the noun itself should be used.

Anyway, is there any mistakes or leakages in those rules?

so I can say that:

a friend of mine = my friend
a computer of yours = your computer
a house if his = his house
a book of hers = her book
a school of ours = our school
a car of theirs = their car
a part of it (not a part of its) = its part
a shirt of Mary's = Mary's shirt (Mary has many shirts)
a shirt of Mary = Mary's shirt (Mary can either has only one shirt or many shirts)
a wing of a bird = a bird's wing
a leg of a robot = a robot's leg (one of the robot's leg)
an office of an engineer = an engineer's office (one of the or only office(s) of a certain engineer)
an enemy of France = France's enemy (one of France's enemies)

Is that correct?

Thank you for all of your answers. They are very helpful.