While I don’t know about lutris specifically, usually the easier strategy is to have your home folder as its own partition and map the new install into your old home folder. Then, once you install your programs again (lutris), it all just maps up properly like you never changed os’.
If you wanted a smaller version of that, backup your lutris configuration and install directories and restore them (in the exact same place) on the new install.
Every app has access to an api that asks your phone to verify your fingerprint against the ones stored for the lockscreen. The phone then asks for your fingerprint and tells the app if it passed or not.
As the commenter pointed out, it’s not an image of your fingerprint either. It’s a hash that is stored on your phone that is likely unique to that device.