What is the purpose to making any sort of distinction between the identity of one person, and the identity of another?

Here is what Parfit had to say:

My life seemed like a glass tunnel, through which I was moving faster every year, and at the end of which there was darkness... [However] When I changed my view, the walls of my glass tunnel disappeared. I now live in the open air. There is still a difference between my life and the lives of other people. But the difference is less. Other people are closer. I am less concerned about the rest of my own life, and more concerned about the lives of others.

This appeals to me, however like you mentioned, on a practical level there might be a desire make distinctions. Your example of forgoing a marshmallow now, so that those like you can have two, is a good example that.