Is the word my a plural pronoun?

No, the pronoun 'my' is singular.

The pronoun 'my' is a possessive adjective, a Word placed before a noun to describe that noun as belonging to the person speaking (one person).

The plural, first person, possessive adjective is 'our'.

Example:

As your mother, you are my responsibility. (singular)

As your parents, you are ourresponsibility. (plural)