My and mine are both called possessive pronouns
My is called a possessive determiner
What is mine also called?
Ron Cowan The Teacher's Grammar of English(Cambridge) calls them possessive determiners and possessive pronouns(Chapter 13). Sorry, I have a paper book.
Possessive determiners are sometimes called possessive adjectives, weak possessive pronouns, genitive pronouns, possessive determiner pronouns, or simply possessives.