Is the word my a possessive pronoun?

No, the pronoun 'my' is a possessive adjective, a Word placed before a noun to describe that noun as belonging to the person speaking.

The corresponding possessive pronoun is 'mine', a Word that takes the place of a noun that belongs to the person speaking.

Example functions:

That's my house on the corner. (possessive adjective, describes the noun 'house')

The house on the corner is mine. (possessive pronoun, takes the place of the noun 'house')