WooCommerce - Saving new "My account" registration fields

functions.php

<?php
/**
 * Save the extra register fields.
 *
 * @param int $customer_id Current customer ID.
 */
function wooc_save_extra_register_fields( $customer_id ) {
    if ( isset( $_POST['billing_first_name'] ) ) {
        // WordPress default first name field.
        update_user_meta( $customer_id, 'first_name', sanitize_text_field( $_POST['billing_first_name'] ) );

        // WooCommerce billing first name.
        update_user_meta( $customer_id, 'billing_first_name', sanitize_text_field( $_POST['billing_first_name'] ) );
    }

    if ( isset( $_POST['billing_last_name'] ) ) {
        // WordPress default last name field.
        update_user_meta( $customer_id, 'last_name', sanitize_text_field( $_POST['billing_last_name'] ) );

        // WooCommerce billing last name.
        update_user_meta( $customer_id, 'billing_last_name', sanitize_text_field( $_POST['billing_last_name'] ) );
    }

    if ( isset( $_POST['billing_phone'] ) ) {
        // WooCommerce billing phone
        update_user_meta( $customer_id, 'billing_phone', sanitize_text_field( $_POST['billing_phone'] ) );
    }
}

add_action( 'woocommerce_created_customer', 'wooc_save_extra_register_fields' );