(I'm not from the UK) It makes sense to me. The Midlands is the middle part of England. It was later divided into the West and East Midlands, so you can live in the west side of East Midlands, as well as live in the east side of West Midlands. I think Derby or Leicester would count as "west side of East Midlands".
Oh you're from NWSEA? Me too! Not from the SW hopefully, everybody there (about 83M people) is a simpleton or a degenerate. I'm from SSE, where it's a completely different story.
It's looks ridiculous now, but all the "north", "south", "midwest", and "west" designations make sense if you think about how they named them all when the US was young and everybody lived on the east coast.
It's also why the "south" stops before even reaching halfway across the southern half of the US.
In the region of Westphalia, there's a part officially called East Westphalia (or Ostwestfalen in German). So you're always wondering, is it in the east or in the west?