I think rank is between the student and teacher. It doesn't necessarily translate anywhere else (other dojo or organizations) and at best represents a process of long, consistent training.
If rank were just between the student and the teacher then why do we need to have special certificates sent from Japan, make a big deal of presenting such certificates, require to people to wear different color belts based on rank, reserve hakama for some ranks and not others, put up a prominent list of members by rank (nafudakake) in the dojo, advertise what rank a teacher has on websites and other promotional material, etc.? Rank actually is a product and tool of organizations that they use to obtain, retain, and control students politically, technically, and financially. Its connection to the student-teacher relationship exists only to the extent that this relationship can be appropriated by the organization to advance its own goals.