Grouping related postmeta data via SQL query

If I understand your question correctly, this should work:

function get_event_info($event_id = 0, $info = '' ) {
  global $wpdb;
  return $wpdb->get_col(
    $wpdb->prepare(
      "SELECT meta_value from $wpdb->postmeta WHERE 
      meta_key = '%s' AND post_id IN
      (SELECT DISTINCT post_id FROM $wpdb->postmeta WHERE meta_key = 'event_id' AND meta_value= %s )",
      $info, $event_id
    )
  );
}

Use this like so:

$names = get_event_info( 19 , 'Name' );
if ( ! empty($names) ) {
  echo '<ul>';
  foreach ($names as $name) {
    echo '<li>' . $name . '</li>';
  }
  echo '</ul>';
}

Please be sure the second parameter is exactly the same as the meta_key: in your example meta_key = 'Name' but the SQL query is WHERE meta_key='name', case is different!