New Menu Item Type Meta Box

Yes, this should certainly be possible, although as you have noticed it’s not too common.

If you take a look at respective admin template the left column is built like this:

<?php do_accordion_sections( 'nav-menus', 'side', null ); ?>

Essentially they are just metaboxes, though differently presented.

The defaults are handled by wp_nav_menu_setup(), which should be a good starting point to taking it apart.

Unfortunately I am not confident to weight in on JavaScript side of it, since that’s not my area.