App Store asks for old Apple ID when installing updates
To delete the application from the Launchpad. Hold down the icon until they all jiggle and the (x) appears. Click that on the app you want to delete, then confirm.
If you do this while the App Store's Updates page is displaying the application, you can then just update!
If not, you will have to download it again. This might mean paying for it again. But if it's family plan eligible, it will show as needing to buy it and you'll have to confirm payment, but then it will inform you that it's family plan eligible and you'll get to download it again for free.
If going through this leaves the app missing from the Launchpad (it did for me one time), then drag the app to the trash (confirming with your password if necessary), wait 10 seconds, and drag it back (possibly confirming again). That should alert Launchpad to its presence.
I hope this helps people in the future. I wasted a lot of time figuring this out, and couldn't find a clue on the web.