Is there any way in the config for a new database name, user and password to be created automatically when a user creates a new site? Currently it uses the same database as the master website.

Yes this is possible but you have to use a MySQL user with "root" privileges.
See the user manual page 14 that speak about the "multisites.cfg.php" configuration file where you can setup the MySQL root login and password.
This MySQL root login is used when a new DB is defined and also used to automaitcally create the MySQL users with its password.

See also the "keywords" that can be used in the setup of a JMS Template rules.
You have some of them that can be used like {user_id}, {rnd_psw}, ...
They can help creating unique DB name, users and passwords.

We added the user with root privileges to "multisites.cfg.php" but now returns the following error:
"JDatabase::getInstance: Could not connect to database
joomla.library:2 - Could not connect to MySQL host=[localhost] with user="