As opposed to the northern/southern hemisphere that has:
- a clear reference point (equators)
- traditionally we don’t model a rotation north/south
Western/eastern hemisphere
- doesn’t have a clear reference (Greenwich?)
- traditionally we model rotation on the north/south axis so western/eastern would depend from the time
Phrases like the the West or Middle East make sense when all relative to Great Britain in geography.
The British had colonies all over and the terms like middle east and the west stuck. It’s also a reason for the most common world map, the Mercator projection.
To add to your comment: The Middle East is called „Naher Osten“ (close east) in German. „Der Osten“ (the east) on the other hand refers to the ex-GDR States.