"I am his younger sister."
の is a linking particle used to modify a noun with another noun. Moving its placement changes what it is being linked to.
AのB - "The B of A" or "A's B"
彼の妹 - "His younger sister", "The younger sister of him" 彼の私 would mean "His me" or "The me of him" and doesn't make sense.
彼の私は妹です "On the topic of his me - a younger sister"
私は彼の妹です "On the topic of me - his younger sister"
The subject here isn't the brother; it is the speaker "I/me" that is being described as having the quality of "being his younger sister"