Why am I receiving a wrong password error when updating apps in the App Store?

In the support article: If you subscribe to Apple One and use a different Apple ID for iCloud - Apple Support Apple alludes to using a different Apple ID just for media, subscriptions, and purchases, but doesn't describe how to do this. See if these work:

In iOS/iPadOS: Go to Settings > [your Apple ID] > Media & Purchases. Sign out of the current ID and sign in with a different one.

In Ventura or later: First, quit Settings. Then, in App Store, choose “Sign Out” in the Store menu, then click on the “sign in” button that appears in the main window. You can then sign into the other account. (from: Using different Apple ID for media & purc… - Apple Community )

Go to Settings > [name on top] > Media & Purchases, and confirm that it is showing the correct Apple ID.