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 will need to reacquire the app with your new Apple ID. That means repurchasing paid apps.
M1 Mac mini, macOS 13.5.2; Watch, series 7 GPS/cellular, watchOS 10.0.1; iPad Pro 2nd Gen, iPadOS 17.0.1; iPhone 12 Pro, iOS 17.0.1; TV 4th Gen, tvOS 17; HomePods, audioOS 17 and Airport Express w/ Airplay 2