But I am not thinking about it from the caregiver's perspective, but rather the object of the narcissism.
It would make logical sense to care for a child you brought into the world because it didn't ask to be born. You are protecting it until it grows, not keeping it forever. When you are raising a child, it gains more and more independence as the years progress. Then it usually leaves the nest and decides whether or not it wants to remain in touch. It is given an opportunity to have independence, and then decides if it still loves the parent enough to come back at holidays.
The same cannot be said of a pet.