difference between a friend of mine and a friend of me

a friend of mine - () one of the friends. If I only have one friend and he is my only friend, we cannot then say "he is a friend of mine", because the "mine" doesn't mean a group of people. If I introduce my daughter I would never say
this is a daughter of mine
a friend of me - ()We (BE) don't say "a friend of me"

Why do you say “friend of mine” instead of “friend of me”?

• In the construction

  friend of mine

  the "mine" means "my friends", so literally

   he is [one] friend of [all my friends]

  or more idiomatically,

  he is one of my friends

  If I only have one friend and he is my only friend, we cannot then say "he is a friend of mine", because the "mine" doesn't mean a group of people. If I introduce my daughter I would never say
  
  

  * this is a daughter of mine,

  always

  this is my daughter. 

a friend of me/a friend of mine

(Is "a friend of me" correct? 

And if it is, what is the difference between "a friend of me" and "a friend of mine"?

I have talked with some none native English speakers those days and they always said "a friend of mine" but as they're not native English speakers, I'm not 100% sure...)

• For whatever reason, we (AE) usually say "a friend of mine."

• 'A friend of mine; a book of yours; a class of his'-- this is how we use the possessive pronouns, Olaf.

• We (BE) don't say "a friend of me".

'a friend of mine' = 'my friend'?

• "X is a friend of mine" means, effectively, "X is one of my friends", the implication being that you have several friends and that X is one of them. In "X is my friend", the implication is that you have only one friend, X.