Below is a guide for taking/preparing/restoring partial backups:
To restore just a single database out of a backup with multiple databases, you basically just restore individual tables. So if you wanted to restore an entire database, you would just restore all the tables in that specific database from the backup. That for restoring individual tables can be found here:
So how you go about it mostly depends on what you are trying to do. If you want to start with a partial backup, then I’d follow the first guide. That will give you a backup of just the DB(s) you want, which you can then restore. If you want to just cherry pick specific databases out of a full backup, then you’ll have to follow the second linked guide to restore individual tables. This process can be difficult if there are lots of tables. You can script it, and there are probably scripts out there if you Google around, but it’s not necessarily easy if you are not very familiar with MySQL and Xtrabackup in general.