In the sentence 'Your brother tied his shoes' is your a noun?

No, the Word 'your' is a pronoun a possessive adjective, a Word placed before a noun to describe that noun as belonging to someone or something.

In the given sentence, the pronoun 'your' tells us the brother 'belongs' to you. His is also a possessive adjective, it tells the shoes belong to him (brother).

The Words 'brother' and 'shoes' are nouns, a Word for a person and a Word for things.