Is the word my a noun?

No, the Word 'my' is a pronoun, a Word that takes the place of a noun in a sentence.

The pronoun 'my' is a possessive adjective, a Word placed before a noun to describe that noun as belonging to someone or something.

The possessive adjectives are: my, your, his, her, its, our, their.

The pronoun 'my' describes the noun as belonging to the person speaking (a first person pronoun).

Example: My mother told me to clean myroom.