Android identifies non-store apps by checking the package installer signature registered during setup. Key identifiers include a missing or non-standard installer package name, lack of an official marketplace signature, and system flags indicating manual APK sideloading. Android installer only sees these installation keys or flags, not what's "inside" the app or any user information. Although Android and Windows app security methods aren't the same, the warning you see is similar to one that a Windows device displays if you are trying to install a program that's incompatible with the device hardware or software that's already installed. If an app isn't sourced and installed from the Play Store, it could interact or otherwise compromise the integrity of the banking app, or any other app that was installed via the Play Store. I had the same issue with a side loaded GPS app that I needed for work with a Chinese contractor. Although I was no longer using the app, I had to remove it as several Play Store apps either wouldn't install or launch due to its presence on my phone. As a workaround, you may need to have two phones: a "clean" one for banking and other apps that require personal information, and another one for the apps that have been side loaded or not available on Play Store.
