" However, I notice that they have "My Reservations", "My Orders", "My Devices", and "My Apple Services" (which includes "Apple Music" and "iCloud"). "
I don't have My Apple Services in there. See:

My Devices are correct, as they're obviously linked to my iCloud Apple ID. But no, the My Orders does NOT show my orders, nor does My Favourites show my favourites either. That's the point of failure within the app login at play.
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What you said below is the EXACT issue we're all having:
" I would say that this is a poor oversight. If the Apple Store App should have its "sign-in information" shared with anything, it should be with the iTunes & App Store.
Even better is to have it be its own. "
iCloud Apple ID can be set, iTunes/App Store Apple ID can be set, but Apple Store Apple ID cannot be set. That's the major problem. And continues to be for the last two months since the last update to the Apple Store iOS app (the major change being introducing Dark Mode to it).
Feedback, mmm... as I said here:
https://forums.macrumors.com/posts/28699722
" NOTE: getsatisfaction.apple.com is a failure – there is literally nowhere on there to report it; all routes lead to "not our department" responses.
The only place I could attempt to report this was inside the Apple Store iOS app, itself!
(your name round symbol (top right corner) > Give Feedback > Take a quick survey) "
As a longterm experienced Apple user, I'm frankly shocked by such a bleeding obvious login oversight.