I try to add a birthdate field in the Woocommerce Checkout form, then save it as a user meta. I can display it with the code below but i can't save it in order to see it in the user profile page.
 add_filter( 'woocommerce_billing_fields', 'add_birth_date_billing_field', 20, 1 );
    function add_birth_date_billing_field($billing_fields) {
    
        $billing_fields['billing_birth_date'] = array(
            'type'        => 'date',
            'label'       => __('Birth date'),
            'class'       => array('form-row-wide'),
            'priority'    => 25,
            'required'    => true,
            'clear'       => true,
        );
        return $billing_fields;
    }
       
    
         add_action( 'woocommerce_checkout_update_customer', 'save_checkout_account_birthday_field', 10, 2 );
            function save_checkout_account_birthday_field( $customer, $data ){
                if ( isset($_POST['billing_birth_date']) && ! empty($_POST['billing_birth_date']) ) {
                     $customer->update_meta_data( 'billing_birth_date', sanitize_text_field($_POST['billing_birth_date']) );
                }
            }
 
                        
Here is the complete code that will display billing birthdate in checkout, in My account Addresses, In admin Orders pages and in WordPress user dashboard:
Code goes in functions.php file of the active child theme (or active theme). Tested and works.