North Coast and South Coast. For example, if you enter with a fleet into Bulgaria from Aegan, your fleet will be positioned on the SC and you will not be able to move to Black Sea directly in the next turn. However, you still have a complete control of the supply center.
Hope, I'm not wrong...

That is correct.

Yes, it's just a division for movement and support purposes.
From the rulebook:
"
Bulgaria, Spain, and St. Petersburg: These are the only coastal provinces that have two separately identified coasts. A Fleet entering one of these provinces enters along one coast and can then move to a province adjacent to that coast only. The Fleet, nevertheless, is considered to be occupying the entire province. Such a Fleet should be placed on the coastline rather than completely inland.

For example, a Fleet at Spain’s North Coast can’t be ordered to move to the Western Mediterranean or to the Gulf of Lyon or to Marseilles. It is, however, considered to be occupying all of Spain. If a Fleet is ordered to one of these provinces and it’s possible for the Fleet to move to either coast, the order must specify which coast, or the Fleet doesn’t move. For example, a Fleet in Constantinople can move to Bulgaria’s East or South Coast. The order would be written “F Con–Bul EC” or “F Con–Bul SC.” Likewise, a Fleet in the Mid- Atlantic Ocean can move to Spain’s North or South Coast, but the order must specify which coast.
"

Hmm, I just noticed that we don't have a East Coast for Bulgaria on our webDiplomacy map. I's called the North Coast instead.