Overwrite menu.twig only for specific node types
Menus do things a little differently then, so you have an option or two...

/**
 * Implementation of hook_theme_suggestions_alter().
 */
function mymodule_theme_suggestions_alter(array &amp;$suggestions, array $variables, $hook) {
  if ($hook == 'menu' &amp;&amp; ($node = \Drupal::routeMatch()-&gt;getParameter('node'))) {
    $content_type = $node-&gt;bundle();
    $suggestions[] = 'menu__node_type__%';
    $suggestions[] = 'menu__node_type__'.$content_type;
  }
}


This will give you give you the following suggestion in this order:


menu--MENU-NAME.html.twig
menu--node-type--page.html.twig
menu--node-type--%.html.twig
menu.html.twig


With this the menu--MENU-NAME template still takes preference over the node type, so we need a trick to add suggestions after.

/**
 * Implementation of hook_theme_suggestions_HOOK_alter().
 */
function mymodule_theme_suggestions_menu_alter(array &amp;$suggestions, array $variables) {
  $key = array_search('menu__'.$variables['menu_name'], $suggestions);
  if ($key !== FALSE &amp;&amp; ($node = \Drupal::routeMatch()-&gt;getParameter('node'))) {
    $content_type = $node-&gt;bundle();
    array_splice($suggestions, ++$key, 0, array('menu__node_type__%', 'menu__node_type__'.$content_type));
  }
}



menu--node-type--page.html.twig
menu--node-type--%.html.twig
menu--main.html.twig
menu.html.twig


Targeting your placement within the array of suggestions can be helpful, but it can be over kill, so you may just want to keep it simple and place your new suggestion at the end of the list and have the highest override

/**
 * Implementation of hook_theme_suggestions_HOOK_alter().
 */
function mymodule_theme_suggestions_menu_alter(array &amp;$suggestions, array $variables) {
  if ($node = \Drupal::routeMatch()-&gt;getParameter('node')) {
    $content_type = $node-&gt;bundle();
    $suggestions[] = 'menu__node_type__%';
    $suggestions[] = 'menu__node_type__'.$content_type;
  }
}


You can even throw $variables['menu_name'] into the mix with:

$suggestions[] = 'menu__'.$variables['menu_name'].'__node_type__%';
$suggestions[] = 'menu__'.$variables['menu_name'].'__node_type__'.$content_type;