I am sorry you can provide the database name in the following variable:
$new->data['softdb'] = 'Database Name';
and the database user will the same as the database name specified.
The database password will be generated by Softaculous which will be 10 random characters.
And for the database host it will take from the server where Softaculous is installed.
For eg: You are using cPanel and the database host used by cPanel is "localhost" then the database host used by Softaculous will be "localhost".
The databases will be separate for each installation and they will not share the same database.