Since menu templates do not have direct access to the user object, we need to pass it manually in my_theme.theme
function my_theme_preprocess_menu(&$variables) {
$current_user = \Drupal::currentUser();
$variables['current_user_name'] = $current_user->getDisplayName();
}
Now, you can use it inside menu.html.twig
<p>Hello, {{ current_user_name }}!</p>