BTW Borough President is pretty much a powerless sinecure now, I therefore see little point for their continued existence.
Eh, I imagine they fill lots of useless patronage jobs, and the GOP likes having Borough Presidents because they can claim Staten Island is still nominally under “Republican rule” and thus they are not completely shut out of city politics.
Far more obscure, the city of Chicago still technically has townships, just like suburban Cook County (even though Chicago township governments were abolished there in 1902), and the townships are still used for property assessment purposes. Nobody in Chicago cites what "township" they are from though, everything is organized by ward or neighborhood (e.g. when's the last time you heard a Chicagoan say they are from "Jefferson Township"?).
Chicago also still has the title of "Vice Mayor", which is supposedly the deputy mayor on paper, but whoever holds the honorary "office" of "Vice Mayor" has even less power than NYC's Borough Presidents, and won't become Mayor if the Mayor dies or resigns.
BTW, I just googled it, the current "Vice Mayor" of Chicago is Alderman Tom Tunney.