An app is forever tied to the Apple ID that acquired it -
An App Store app is forever tied to the Apple ID used to acquire it (paid or free). The Apple ID is encoded into the licensed copy that is downloaded. It is in the copy's DNA. That is the only Apple ID that can redownload or update the app. If you cannot reclaim the old Apple ID, you must reacquire the app with your new one. That means repurchasing paid apps.
M1 Mac mini, macOS 13.2.1; Watch, series 7 GPS/cellular, watchOS 9.3.1; iPad Pro 2nd Gen, iPadOS 16.3.1; iPhone 12 Pro, iOS 16.3.1; TV 4th Gen, tvOS 16.3.2; HomePods, audioOS 16.3.3 and Airport Express w/ Airplay 2