In short, there are enough words out there that we could all have completely unique names. In combination, assuredly, but a not insignificant chunk of us might be single-worders, like Cher and Madonna and Putt and Menthol. And Shampoo. And Skelter. And Hazmat.
I'll note that I would not want to be named Venous Thrombosis, but when I was 9 and looking up interesting terms in the encyclopedia to bestow upon my puppy Ginger as her registered name, it looked pretty good. Until my mom told me that was Utterly Ridiculous. She ultimately picked Sweet Singabera--singabera being the Sanskrit word for ginger. (And you wonder where I get it? No, of course not...) In any case, I wouldn't want to be named Venous Thrombosis, but I might accept it if I were the ONLY Venous Thrombosis on earth.
Anyway. I suspect that single-word names would be highly prized, so there'd have to be a randomizer (that then takes names out of the collective pot when people are named, and add names back in a certain number of years after people die--if they don't have a Fame Factor greater than 5) to ensure that single-word names don't go to the wealthy or the privileged or whatever. Plus, that's how I, Venous Thrombosis, would become friends with Peanut Butter Effulgence, or Geosynchronous Pasties.
I suspect, now that I've put in almost ten whole minutes of thought into this, I might use this background for some screwball comedy of the distant future.
In the meantime, call me Vene.