When you are going to make people's first name into possession, remember that's only possible for two position:
1. Kindness. For example: امیرم means you have a close relationship with Amir as if he is yours, or parts of your existence!!
2. Family hierarchy: For example: Amir is Maryam's son. So when Maryam says امیرم it means Amir belongs to her as family member and again it's a form of kindness. (Almost all the time, it is limited to first order family (Father, Mom, son, daughter) or couples.