If you pass the ID to wp_nav_menu()
like this:
wp_nav_menu( array(
'menu_id' => 'foobar-nav',
) );
Then you can access it through the $args
parameter:
function start_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
$menu_id = $args['menu_id'];
}