Were those apps installed originally using the Apple ID that you currently have associated with the App Store? For example, I started fresh with my iPhone 6 and ended up with my iCloud Apple ID in the App Store. However, I use a separate Apple ID for App Store purchases. While my iCloud Apple ID was still there, I installed several apps and then realized and changed over to my other Apple ID in the App Store settings. After that, I would get bugged about entering the password for my iCloud Apple ID for updating those apps installed under it. I had to delete and install them again with the Apple ID I wanted to be using from the start.
Is this a completely random Apple ID you don't even know that it's asking you to enter the password for, or is it possibly one that was used at one time to install that app originally? I would simply delete the apps you think are causing this issue and install them with your Apple ID in the App Store settings. I was able to find out which apps were causing this issue for me when I backed up my phone + apps through iTunes. It gave me a warning that it couldn't back up three or four of them, and it clicked right away that those were the apps I had installed with the wrong Apple ID.
Nope! Always only used one ID for everything. And it all worked 100% before I got my 6+ (previously used a 5 on ios7)
The ID is my sister's who I never ever see, and the only connection I have to her is her I had to create a backup of her phone on my laptop when giving her a 4S, but I don't see how that could interfere. The apps were all transferred from my iPhone 5 and then onto this with a restore from my 5's backup, so it's gotta be a glitch.

It doesn't ask me for her password if my account is logged out, either; it only asks me for her password if I'm logged in.