'My' is a...
'my' cannot be a pronoun because it cannot take the place of a noun, it has to precede a noun. e.g. we cannot substitute 'my' for the noun phrase in this sentence: This is the house that belongs to me. We can only substitute 'mine', so 'mine' is the possessive pronoun.
'My' cannot stand alone. It needs to modify something. You can't have 'My is a good one' or 'Look at my!' 'My' need to modify something like house, or shoe, or mitten. In that situation it would determine which shoe or mitten and also indicate possession.
I disagree with Sue. "My" regularly takes the place of a noun in first-person forms of sentences:
1. John's house.
2. My house.
The term "I" is the archetypal first-person pronoun - but presumably Sue thinks this is not a pronoun either because I can find examples of sentences where we cannot substitute "I" for the noun, e.g.:
This is the house that belongs to John.
Whether a word can be uniformly substituted into the noun position in every sentence containing a noun is not a test for whether it is a noun.