I think you don’t quite understand what a ‘database’ is.
You will certainly need to keep player information - their score and progress is important. This isn’t a ‘database’, it’s just information. You can store that in a class and save it any way you like - a text file, XML, upload it to the web etc.
There doesn’t seem to be any reason to use a real database, unless you are storing all of the scores and progress in a central repository (such as a website) so players can compare their scores.