I think it has to be population across all boroughs. No way ancient Rome is more densely populated than Manhattan today. I read the statement on one of the LinkNYC billboards.
The Aurelian Walls contain an area of 13.7 square kilometers, peak population within this area is estimated on the high end at 1m, at that population the density is 73k/sqkm compared to Manhattans 28k/sqkm all New York's boroughs are 10947/sqkm bringing a more accurate total to 6.63x more dense with Manhattan being about 1/3 as dense.
Source: Maths & [The Cambridge Companion to Ancient Rome Population of Rome] (https://books.google.com/books?id=yaM0AAAAQBAJ&printsec=frontcover&dq=The%20Cambridge%20Companion%20to%20Ancient%20Rome%20population%20of%20rome&hl=en&sa=X&ved=0ahUKEwjKt4ik9JvPAhUL9IMKHfJoAIgQ6AEIHjAA#v=onepage&q&f=false)