I tried the following.
1. log out of MAS, quit, log in again.
2. try update the app: same error.
1. delete app from apps folder
2. app store offers "INSTALL" rather than "update"
3. Press "install"
4. Downloads and installs.
5. "Update" button highlights
6. Press "update" - same error: "Update this app from the account you bought it with" - or whatever the wording is.
Ok, here's the latest update on this problem.
I tried to open an app that had been downloaded and I tried to do an in-app purchase (specifically, the FreeMemory app upgrade). Since it was a new hard disk, the computer asked me to authenticate the app against the app store; in other words, the anti-piracy thing kicked in. Then, when I gave my MAS password, the MAS asked me to verify my billing details. Thereafter, it allowed me not only to run the app and do the in-app purchase, BUT it stopped complaining about updates not being permitted with this account.
So maybe this is a bug Apple needs to pay attention to: viz., if the hard disk is new and you think the app store details are faked, ask the user to verify billing details straight away rather than leaving them in the dark as to why they think they're logged in, but they're not authenticated fully, and wondering why they can't update.
I will monitor the situation, and if any further problems come up, I will report them here, but for now, this seems to have fixed it.