App Store products can only be updated by signing in to the store with the same Apple ID that was originally used to download them. You can't update anything, including your own purchases, as long as there are pending updates linked to the other ID.
If you can't sign in with the other ID, you have to delete all the apps that are linked to it. If you don't know which apps those are, see below.
1. Triple-click anywhere in the line of text below on this page to select it:
kMDItemAppStoreHasReceipt=1
Copy the selected text to the Clipboard by pressing the key combination command-C.
2. In the Finder, press command-F to open a search window, or select
File ▹ Find
from the menu bar. In the search window, select
Search: This Mac
from the row of tokens below the toolbar. Below that is a popup menu of search criteria, initially showing Kind. From that menu, select
Other...
A sheet will drop down. In that sheet, select
Raw Query
as the criterion, then click OK or press return.
Now there will be a text box to the right of the menu of search criteria. That's where you enter the raw search query. Click in that box and paste the text you copied earlier by pressing command-V.
3. The search window will now show all the App Store products that are installed. Compare those search results with the list of your purchases from the App Store. To see the complete list, you may need to unhide hidden purchases. If any apps were download from the App Store using other Apple ID accounts that you control, sign in to the store under each of those ID's and check the purchases.
4. At least one of the apps in the Spotlight search results is not among your purchases in the App Store. Delete each such item by following the instructions in the Apple Support article linked above.
5. Quit and relaunch the App Store.
If you find these instructions confusing, ask for an alternative method.
