Sorry for not answering earlier. It is possible to have two addresses, but not for the same object in OSM. I would use the primary and official address for the building and maybe add an address node (for the secondary) within the building. In case the secondary address could be defined to a certain entrance (obviously not in this case) then I would add a node with a entrance=* tag on the building outline and provide it with the secondary address. In any case I would also explain/document the situation to other mapper in the note-tag.