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.

It won't 'forget' the old ID as each download from the Mac App Store embeds the purchasing account. Those previous apps were purchased/downloaded for free on that ID & are only licensed to that ID.

The only way round is to either use that ID for those apps, delete and redownload each affected app one by one or to wipe the machine & start over being careful to not copy back apps from backup copies and only to use the new account in the store.

For corporate use, it would be far better to not purchase through the App Store, but to buy multi-seat licenses directly from the app creators.