I'd assume they would have a record of the user in their database. How else would they earn money from the data harvesting their app is a cover for? For the small number of apps that aren't solely data harvesting, surely they still have records of their users in a database as well? This question almost reads as if you're assuming the only record of an app would be through the store and not by the app developers themselves. I would find that truly shocking and quite comically sad if true.
It seems like it would be trivial for a user to login to the app acquired from a different store to be able to display a "welcome back" or even something along the lines of a "restore purchases" type of thing.
This can't be reinventing the wheel kind of a thing.
Using android.content.SharedPreferences, set one singaling the user purchased a license from the Amazon copy of the app, read it from the Google Play app.
It is doable, the main issues is:
1. Getting users to redownload the app from the Play store
2. Maintaining this registration transfer mechanism
That's on the dev of the app. They should absolutely have the ability to see that you download the app, and allow full access again.
This isn't any different from a user switching to a new device and having to download apps again. Not everyone restores from backup and prefers clean installs. Downloading an app again should not be the point of friction
It seems like it would be trivial for a user to login to the app acquired from a different store to be able to display a "welcome back" or even something along the lines of a "restore purchases" type of thing.
This can't be reinventing the wheel kind of a thing.