As someone who lives in Manhattan, there are many buildings that have entrances on 2-3 different streets: having an address that specifies where the entrance actually is is a feature.
That sounds like something that's solved by a street name & number?
Buildings generally only have one address, usually considered to be wherever the "main entrance" to the building is AFAIK. Additional entrances to the same building could probably be identified by the street they're on, but they won't have been assigned a street number.