Sounds like an issue with the app. Usually, apps have a 'redeem' button somewhere, that checks with the App Store to see if you did the purchase before, exactly

Sounds like an issue with the app. Usually, apps have a 'redeem' button somewhere, that checks with the App Store to see if you did the purchase before, exactly for this scenario. (e.g. Monument Valley does this, and also some apps on macOS.) I believe you can link maybe 5 devices to one account?