Hi,
My test showed that the list of groups (post types) at the “Add menu items” section does not depend from the ability of user to edit those items. My test user does not have capabilities to edit Woocommerce products but he see “Products” group at the mentioned section.
I may just suppose that another plugin may be involved. Did you try to deactivate all plugins and re-test? If that will change a picture, activating plugins back one by one you can isolate a reason.
