Is this custom post type visible at admin menu (wp wp-admin)?
‘Pages’ is a WordPress built-in post type. Your custom post type should be visible at menu editor page as a separate widget, like ‘Projects’ at your screenshot.
OK. I looked at the WordPress source code.
I don’t see any access conditions for this sections at nav-menu. They are the usual meta boxes registered for this page.
Look at the “Screen Options” at the top of the nav-menus.php page, just in case if a “Projects” checkbox is not turned ON for this user.