This building is a modern homage (built 1983) of the demolished Bogardus Building/Edgar Laing Stores (1849-1971), but it's not an exact replica nor at the same location. 
The South Street Seaport Museum lists this as "New Borgadus Building", and a few other online sources call this the "Borgadus Building".
There isn't any signage giving this name from images though, so I feel like this may be more of a local name than an official name?