Adding a custom taxonomy to “nav_menu_item”

This is not how you enable adding your taxonomy terms to menus. To do that you just set show_in_nav_menus to true when registering the taxonomy. You have it set to false.

$args = array(
    'labels'                     => $labels,
    'hierarchical'               => true,
    'public'                     => true,
    'show_ui'                    => true,
    'show_admin_column'          => true,
    'show_in_nav_menus'          => true, // This
    'show_tagcloud'              => false,
    'rewrite'                    => $rewrite,
    'show_in_rest'               => true,
);

register_taxonomy(
    'my_region_taxonomy',
    array(
        'post',
        'page', // Not here
    ),
    $args
);

Your taxonomy should not actually be a taxonomy for the nav_menu_item post type.