Hi! There's typically only ever one database – if you're using the built-in configuration, that's either SQLite (default, stored as a file on disk) or MySQL or PostgreSQL. By default, SQLite writes to a file prodigy.db in your Prodigy home directory (.prodigy in your user home, unless you change it). But you can also customise the location and file name.
If you mean the datasets in the database, you can use the prodigy stats command with -l set to list all datasets present in the database. It will also show you the location of your Prodigy home directory that contains the database file. To export datasets as JSONL, you can run db-out.
If you want, you can also interact with the database programmatically – see here for details: