Guys I don't know if any of this will help, but there are a few theories I can debunk.
1. It's not regional, I'm in Australia and mine just started doing this same thing at random. No idea why.
2. It's not necessarily an old account, I've only ever used one, so it can't be looking for an old one.
3. It's not the password thing, because mine definitely has both upper and lower case and at least a number.
* I thought for a short while that maybe I was getting my password wrong, but it appears this was not the case (see below).
* resetting my device didn't seem to make a difference, nor did logging out of then back into iCloud.
So, I too went clicking things at random to try and sort out the issue and seem to have (at least for now) come up with a solution. This is what worked for me:
Settings -> iTunes and App Stores -> click your Apple ID at the top -> click view Apple ID and enter password (at this point if it logs in, you know your pw is right). Then I clicked on something down the bottom like "remove this device from iCloud" or something similar.
I don't know how this is different from the settings within iCloud but I know they are. Previously when I changed ANY setting in iCloud, it asked me for my pw. After making this change, it went back to its usual behaviour, and any time I turned an option off it asked me if I wanted to keep the data on the iPad, which it definitely wasn't doing before.
As far as I can tell I've corrected my issue.
However I must say, it is Apple, so who knows. I've found in the past that different people on different devices can seem to have the same issue, caused by very different problems.