I don't think that's quite the right interpretation. Let's say that the human population increased by a factor of 1000. Does that mean you have 10 more bits of

I don't think that's quite the right interpretation. Let's say that the human population increased by a factor of 1000. Does that mean you have 10 more bits of uniqueness? You didn't change at all. In actuality, you already had those 10 more bits of uniqueness, and people just need to ask more questions to find out which one of the many unique people you are. This is about identification, not uniqueness.