Bom, estudei vários códigos e cheguei em um. Vou deixar o código abaixo para ajudar outras pessoas:
// Libera editar telefone e celular
add_action( 'woocommerce_edit_account_form', 'add_billing_phone_to_edit_account_form' );
function add_billing_phone_to_edit_account_form() {
$user = wp_get_current_user();
?>
<p class="woocommerce-form-row woocommerce-form-row--wide form-row form-row-wide">
<label for="billing_phone"><?php _e( 'Celular', 'woocommerce' ); ?> <span class="required">*</span></label>
<input type="text" class="woocommerce-Input woocommerce-Input--phone input-text" name="billing_phone" id="billing_phone" value="<?php echo esc_attr( $user->billing_phone ); ?>" />
</p>
<?php
}
add_action( 'woocommerce_save_account_details_errors','billing_phone_field_validation', 20, 1 );
function billing_phone_field_validation( $args ){
if ( isset($_POST['billing__phone']) && empty($_POST['billing_phone']) )
$args->add( 'error', __( 'Digite seu telefone/celular', 'woocommerce' ),'');
}
add_action( 'woocommerce_save_account_details', 'my_account_saving_billing_phone', 20, 1 );
function my_account_saving_billing_phone( $user_id ) {
if( isset($_POST['billing_phone']) && ! empty($_POST['billing_phone']) )
update_user_meta( $user_id, 'billing_phone', sanitize_text_field($_POST['billing_phone']) );
}
Vale lembrar que: billing_phone é para Telefone, billing_mobile_phone é para Celular (caso precise usar telefone/celular no cadastro).