There appears to be no answer to this. Apple won't explain why it requests the password twice; they won't explain why they don't store it in Settings; and other people simply reply stupid things like, "Are you sure you typed it correctly?" and "Restart your iPad." Duh. Apple asks for the password twice for EACH AND EVERY Apple thing you try to do: App Store, iTunes, iCloud -- you name it, even when you have it in your Settings. Can someone PLEASE explain why this happens and why Apple is the ONLY company that does this? (It happens on my iPad Pro and iPhone 5s, iOS 10, even when Touch ID is enabled. Why even have this feature if you have to type the password, twice, anyway? How is this helpful?)
And, no, Tapping Settings>iTunes & App Store, then restarting does NOT fix this. As soon as I try to buy something (song, app, you name it), the ID box pops up again, asks for my password, I type it in, it goes away, then pops back up, asking for my password, which i type again -- exactly the same way -- then goes away, then I can buy it.
It's especially annoying when it happens on my Mac. Isn't that why Apple created the Keychain app? Why do that when it doesn't work.
Here are the details on my Mac, since no Apple people respond unless you include it:
OS X El Capitan, 10.11.6; Mac mini (Mid 2011); 2.3 Ghz Intel Core i5; 8GB 1333 Mhz DDR3
iTunes 12.5.1.21