It's really no different from any other band+song+player+score+date database.
You'll end up with a primary key of (band+song+player) and a secondary key of (player), so with only two keys in the table, any database engine should work fine. (Postgres is notoriously bad at too many keys per table.) Just make sure you have high performance storage and sufficient RAM in your database engine.
You can probably stuff them all in a single SQL database table, unless your game is super popular. At that point, physically shard by some key, or throw it all at Amazon DynamoDB or Google Firebase and pay them to manage it. Contrary to some popular belief, I don't think self-hosted NoSQL solutions like Redis or Scylla or Mongo would be any better for this use case. They're all bound on I/O performance and indexing, and the workload is exceedingly regular, perfectly mapping to a SQL table.
There's also the question of how you map band + song name to a particular score, if the spelling isn't consistent (as when users encode their own files from CDs and such.) ContentID or other content fingerprint services may help you come up with a reasonable key for each track, then.