Multiple addresses for same object

Simplest way: Don't add any addr:* tags to the building itself. Instead, create nodes inside the building perimeter (preferably near the entrances that corresponds to each address) and add the addr:* tags to those nodes. The addresses can be entirely different -- different housenumber, different street.

More complicated: Similar, but instead of address nodes inside the building's perimeter, put entrance nodes directly on the building's perimeter. Tag these with the addr:* info plus add entrance=main or entrance=yes. Possible difficulty -- you might not consider every address to be a true "entrance" to the building. (If it's just the address of a shop on the ground floor of a large building, should that be considered an entrance to the building? Opinions differ on this.)

There are other approaches too, depending on the circumstances. If you show the exact situation I can tell you what I would do ... but ultimately it depends on local consensus. Examine nearby areas that are well mapped, and see how similar situations have been handled. Don't be shy about contacting the local mappers who dealt with these same problems in the past -- sometimes they're long gone, but often they're happy to advise & help.