If we could see all the intersections, down to the point of very small groups (5-10 people), what would it look like?

If each category had two values uniformly distributed, then 29 categories would be more than enough to categorize each American individually.

How many categories non-uniformly distributed will it take?

A visualization for a few categories: https://cran.r-project.org/web/packages/nVennR/vignettes/nVennR.html

We would need something different for higher dimensions!

The data to answer this question might not exist.