How does this controversy relate to the development of the Byzantine Empire?

I believe that "this controversy" is the controversy of naming a school created from a school split: half the school will go to a new institution.

Byzantine empire was similar in that it was created after the split of the Roman Empire. The Byzantine empire choose to keep the old name: they always called themselves "Romans" - but history gave them a different name, to differentiate them from the "other" roman empire.