#project-application

• The following points are just a start and don't necessarily encompass all of the changes that may be necessary
• A specific point may just be an example and may apply in other places
• A review is about code that does not follow the coding standards, contains possible security issue, or does not correctly use the Drupal API
• The single review points are not ordered, not even by importance

src/Controller/AjaxCartUpdateController.php

Since that class does not use methods from the parent class, it does not need to use ControllerBase as parent class. Controllers do not need to have a parent class; as long as they implement \Drupal\Core\DependencyInjection\ContainerInjectionInterface, they are fine.

          $response['cart_block_quantity'][] = $order_item->getQuantity() . ' ' . t('x');
          $response['cart_block_title'][] = $order_item->getTitle();

The translatable string should use placeholders, which would avoid concatenating a translatable string with a string. It would also give to translator people more context.
The code should also use $this->t(), since it is not in a static method.

src/Form/AjaxCartUpdateSettingsForm.php

  /**
   * The typed config manager.
   *
   * @var \Drupal\Core\Config\TypedConfigManagerInterface
   */
  protected $typedConfigManager;

  /**
   * Constructs a new AjaxCartUpdateSettingsForm.
   *
   * @param \Drupal\Core\Config\ConfigFactoryInterface $config_factory
   *   The config factory.
   * @param \Drupal\Core\Config\TypedConfigManagerInterface $typed_config_manager
   *   The typed config manager.
   */
  public function __construct(ConfigFactoryInterface $config_factory, TypedConfigManagerInterface $typed_config_manager) {
    parent::__construct($config_factory);
    $this->typedConfigManager = $typed_config_manager;
  }

$typed_config_manager is passed to the parent class, which already has a property for that.

ajax_cart_update.module

  // Attach translations to drupalSettings.
  $translations = [
    'items' => new TranslatableMarkup('items', [], ['context' => 'Cart block item count']),
    'x' => new TranslatableMarkup('x', [], ['context' => 'Cart block quantity separator']),
  ];

  $attachments['#attached']['drupalSettings']['ajaxCartUpdate']['translations'] = $translations;

JavaScript code that needs translatable strings uses Drupal.t().