Get current post’s nav menu name (term name)

You are missing a link. The posts are not assigned to the menus directly. There is a post type nav_menu_item in nav_menu taxonomy which links posts (or other kinds of destinations) to their place in menu.

This snippet should get you started on retrieval:

$menu_items = get_posts( [
    'post_type'  => 'nav_menu_item',
    'meta_query' => [
        [
            'key'   => '_menu_item_object_id',
            'value' => get_the_ID(),
        ]
    ],
] );

foreach ( $menu_items as $menu_item ) {
    var_dump( get_the_terms( $menu_item->ID, 'nav_menu' ) );
}