Sigh....here is the answer.
If you have an update in the App Store that requires signing into an account that's not yours, it's because there's an app on your machine that has, in its Contents, a special file that is causing the problem.
For example, on my machine, I discovered a copy of Final Cut Pro left over from a former user of my old laptop, that was causing the problem. The old laptop was running Snow Leopard and so I never saw this problem.
If you don't need the app anymore, simply move it to the trash, empty the trash, then reboot. That solved the problem for me.
If you want to KEEP the app, here's a way somebody suggested to fix the problem. Since the first solution worked for me, I didn't have to do this so it's not verified:
1. In the App Store, purchase the Twitter App.
2. In the Finder, right-click on the Twitter app, and choose Show Package Contents.
3. Choose Contents. There should be a file in there called _MASReciept. Copy it.
4. Right-click on the app that you suspect is causing the problem, and choose Show Package Contents.
5. Choose Contents, and in that folder, Paste the new _MASReceipt file over the old one.
Cheers 🙂