From your question, I am not sure if you want an answer only for the phrase “a friend of mine”, or if you are looking for possessive phrases in general. So here are the definitions for partitive and the various genitives that may apply to your situation.
I hope this clears up any confusion.
A partitive does imply ”a distinct quantity that is part of a larger whole”, but the usual prepositions for partitives are some of or any of.
Jenny ate some of the strawberries.
A genitive is the possessive case (in English), and is marked either with the suffix ’s or with an of phrase after a noun.
Jenny is a friend of mine.
A double genitive is indicated by a noun, the preposition of followed by a possessive noun.
Jenny likes Tommy, who’s a friend of Karen’s.
A group genitive is the possessive cases with a misdirection in the possession because the suffix 's appears at the end of a noun phrase where the final word in that noun phrase is not the head (or only head) of the larger phrase.
Jenny is allergic to the man next door’s dog.
An independent genitive is a construct where the noun following the possessive is omitted.
Jenny went to the doctor’s after she fell down the stairs.