I was able to find the solution, and the reason for the issue, in a response on the Apple Support forums:
Quote:
The App Store is scanning the Mac and finding purchased apps in the Applications folder. It then checks against the list and finds that they are old versions and prompts for an update.
All that is required to install an app is a Apple ID and admin rights on the target mac. The apps she is installing are wrapped in DRM which identify her as the owner, but the App Store doesn't care, as she is entitled to install them on your Mac. But it reads her ID from the wrapper and thus prompts for her password to do the update.
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I had no idea it would be scanning for other downloads. Armed with that, I went looking (for this, Spotlight actually worked). I found where we had copied her hard drive from her laptop onto my computer in preparation for some repair work, and every one of the named applications was there. 101.7GB deleted later and without any need to reinstall the App Store was happy to accept me, and let me download programs.
Thanks for the assistance. I truly do appreciate it.