HyperDB, Multisite – on new site creation put site’s tables to specific database

The plugin you are using hasn’t been updated for two years. It is better not to rely on plugins so old. You should consider writing an sql script, that would create a database, as per the MySQL documentation. And simply run the SQL command through php. You can check how that can be done in here:
https://www.w3schools.com/PHP/php_mysql_create.asp

After that if your databases are WordPress databases and you insist on using the wpdb object, you must initiate a new wpdb object. Like so:

$custom_wpdb_object = new wpdb( $dbuser, $dbpassword, $dbname, $dbhost );

And use it how it was intended.