How to save wordpress username in woocommerce client billing address?

Question

WordPress

Good afternoon.
Added the ability to enter a name when registering a profile in wordpress. I want the name to be saved in two places, in the username and in the name that is indicated on the checkout page in woocommerce.
Here is the code I found on the web and it allowed me to add a name to the wordpress registration page:
//1. Adding a new element to the form
add_action( 'register_form', 'myplugin_register_form' );
function myplugin_register_form() {
$first_name = ( ! empty( $_POST['first_name'] ) ) ? trim( $_POST['first_name'] ) : '';
}
//2. Add a check if this element is required
add_filter( 'registration_errors', 'myplugin_registration_errors', 10, 3 );
function myplugin_registration_errors( $errors, $sanitized_user_login, $user_email ) {
if ( empty( $_POST['first_name'] ) || ! empty( $_POST['first_name'] ) && trim( $_POST['first_name'] ) = = '' ) {
$errors->add( 'first_name_error', __( 'ERROR: You must include a first name.', 'mydomain' ) );
}
return $errors;
}
//3. Store the new field in the user's metadata
add_action( 'user_register', 'myplugin_user_register' );
function myplugin_user_register( $user_id ) {
if ( ! empty( $_POST['first_name'] ) ) {
update_user_meta( $user_id, 'first_name', trim( $_POST['first_name'] ) );
What should be added here and where exactly so that the entered name during registration is also saved in the billing address section of the woocommerce client?