Are you sure about that? Look at F-Droid or NewPipe. Both are third-party, and both store settings and alike without losing stuff.
I have actually located the directory of another third-party app, Mitch, an Android itch.io client. Preferences are stored as expected in an XML file that is located in /data/data/<package_name>.
So it should be entirely possible to save a game on Android. Yes, things have changed, but that doesn't break them.