woocommerce_checkout_update_order_meta action is not working

I have just change a little bit your last hooked function and it works (on WC version 2.6.x and 3.0+). It's better with empty() php function to use variables (to be retro compatible).
Also is better to use update_post_meta() instead of add_post_meta() as this function will make sure that the meta_key already exists and if not, add_post_meta() will be called instead...

Here a screenshot of the wp_postmeta table related to the order meta data:

If the meta_key don't start by an underscore like here, it appears in backend order edit page in the Custom fields metabox:

Here is this code:

add_action( 'woocommerce_checkout_update_order_meta', 'saving_checkout_cf_data');function saving_checkout_cf_data( $order_id ) {    $recipient_address = $_POST['recipient_address'];    if ( ! empty( $recipient_address ) )        update_post_meta( $order_id, 'recipient_address', sanitize_text_field( $recipient_address ) );    $recipient_phone_number = $_POST['recipient_phone_number'];    if ( ! empty( $recipient_phone_number ) )        update_post_meta($order_id, 'recipient_phone_number', sanitize_text_field( $recipient_phone_number ) );}

Code goes in function.php file of your active child theme (or theme) or also in any plugin file.

If you want to have a meta_key starting by _billing… like classic billing checkout fields, you just need to change that in update_post_meta() function. For example:

update_post_meta( $order_id, '_billing_recipient_address', sanitize_text_field( $recipient_address ) );

But in this case, this will not appear in the custom Fields metabox in the order edit page.