I’ve asked this question a bit to early, but to help others in the feature here’s the solution.
For the menu editor to appear, you need to register a menu first:
add_action( 'init', 'register_header_menu' );
function register_header_menu() {
register_nav_menu('header-menu',__( 'Header Menu' ));
}
Or as pointed out by Howdy_McGee:
add_theme_support( 'menus' );