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I think the case is homemade, and perhaps the board is too. I can't say anything about the pieces because I would need to see a close-up side or oblique view, in fact several views. A closer view of the board and case would be needed too.
Regarding the board's 90 degree rotation in the case, I've only seen this arrangement for sets where the lid is removable, See the French set in my initial post as an example. Likely, your set was assembled incorrectly if it was meant for play between two players. On the other hand, if it is an analysis set it doesn't matter.
I hope this helps.
I didn't realize your set came with discs too; obviously, I just skimmed the description at ChessMate.
I understand and concur with your distinction between planes, trains and automobiles, so to speak, and your exam room. The ChessMate looks good and fits quite well where you have it; it's neither too large nor too small, whereas the "wings" of a Whittington would be in the way.
Do you ever have a patient who wants to play while your examining his eyes?
Q: Please read the bottom row of characters; A: d4d5c4 (ha ha).
Actually a few kids from the scholastic club I run come to me for their vision care. One boy in particular has wanted to play after his exam and while his parents were finalizing the glasses purchase.
It must be fun or why do it at all.