Drupal - How to add module to admin/config page?

If you do it like this, with the system.module callback for the parent item, you get the nice listing page when you visit 'admin/config/mymodule'

/**
 * Implements hook_menu().
 */
function MYMODULE_menu() {

  $items = [];

  $items['admin/config/mymodule'] = [
    'title'            => 'My configuration section',
    'description'      => 'This is the parent item',
    'position'         => 'left',
    'weight'           => -100,
    'page callback'    => 'system_admin_menu_block_page',
    'access arguments' => ['administer site configuration'],
    'file'             => 'system.admin.inc',
    'file path'        => drupal_get_path('module', 'system'),
  ];

  // Need at least one child item before your section will appear.
  $items['admin/config/mymodule/item'] = [
    'title'            => 'First item',
    'description'      => 'This is the first child item in the section',
    'page callback'    => 'mymodule_item_callback',
    'access arguments' => ['administer site configuration'],
  ];

  return $items;
}

Looking at the code of system_admin_config_page(), which is page callback for admin/config, I notice it contains the following lines:

if ($admin = db_query("SELECT menu_name, mlid FROM {menu_links} WHERE link_path = 'admin/config' AND module = 'system'")->fetchAssoc()) {
  $result = db_query("
    SELECT m.*, ml.*
    FROM {menu_links} ml
    INNER JOIN {menu_router} m ON ml.router_path = m.path
    WHERE ml.link_path != 'admin/help' AND menu_name = :menu_name AND ml.plid = :mlid AND hidden = 0", $admin, array('fetch' => PDO::FETCH_ASSOC));
  foreach ($result as $item) {
    _menu_link_translate($item);
    if (!$item['access']) {
      continue;
    }
    // ...
  }
  // ...
}

The first query select the menu_name field for the menu associated with the path admin/config, which by default is management; the second query select all the menus that have the same value for menu_name, and whose parent is admin/config.

As your menu uses a different value for menu_name, it is not selected from the second query, and it is not show in the admin/config page.

Remove the 'menu_name' part, that is not necessary.