How to drop a database from libreoffice?

I need to drop / de-register test/trial databases from libreoffice.

Easy. Just choose MENU Options (Mac OS Preferences) > LO Base > Databases and delete your trial db from the Registered databases list.

How could I have missed that! However, all the databases are not listed therein. Has the bibliography database listed there anything to do with it?

Well, the DB registration list is arbitrary and you can add or subtract from it at will. Your new DBs do NOT necessarily get added by default. (Not sure if the bibliography DB is added by default, but you should be able to add it if needed.)