The function metaDatabases() returns a list of the databases visible to the connected user. If the driver does not support the command then it returns false. Visibility of a database does not assume any rights are available to the connected user.

/*
 * Connection to MySQL database assumed
 */
$d = $db->metaDatabases();
print_r($d);
/*
 * returns Array
          (
          [0] => information_schema
          [1] => performance_schema
          [2] => piwik
          [3] => test
          [4] => world
)
*/