EDIT

For those wondering how to do the same thing, look into what the accepted response mentions about altering the route, because you basically need to overwrite how flag does its job and add your code as an extension of it.

So, I have a RouteSubscriber.php file with the following

namespace Drupal\my_module\Routing;

  use Drupal\Core\Routing\RouteSubscriberBase;
  use Symfony\Component\Routing\RouteCollection;

  /**
   * Listens to the dynamic route events.
   */
  class RouteSubscriber extends RouteSubscriberBase {

    /**
     * {@inheritdoc}
     */
    protected function alterRoutes(RouteCollection $collection) {
      // We need to take control of the default methods for the Flag controller so that we can hook into it's actions and add our own
      if ($route = $collection->get('flag.action_link_flag')) {
        $route->setDefault('_controller', 'Drupal\my_module\Controller\FlagController::flag');
      }
      if ($route = $collection->get('flag.action_link_unflag')) {
        $route->setDefault('_controller', 'Drupal\my_module\Controller\FlagController::unflag');
      }
    }
  }

Then I have a FlagController.php with code such as

namespace Drupal\my_module\Controller;

use Drupal\Core\Ajax\AjaxResponse;
use Drupal\Component\Plugin\Exception\InvalidPluginDefinitionException;
use Drupal\Component\Plugin\Exception\PluginNotFoundException;
use Drupal\flag\FlagInterface;
use Drupal\flag\Controller\ActionLinkController;

/**
 * Controller responses to flag and unflag action links.
 *
 * The response is a set of AJAX commands to update the
 * link in the page.
 */
class FlagController extends ActionLinkController {
  /**
   * Performs a flagging when called via a route.
   *
   * @param FlagInterface $flag
   *   The flag entity.
   * @param int $entity_id
   *   The flaggable entity ID.
   *
   * @return AjaxResponse|null
   *   The response object, only if successful.
   **/
  public function flag(FlagInterface $flag, $entity_id) {
    $response = parent::flag($flag, $entity_id);
    // The line above means your code still does what Flag needs to do, so you need to keep this in, or you'll stop flag working
    
    // Now you can add your own commands based on this link, like below
    https://www.drupal.org/docs/drupal-apis/ajax-api/core-ajax-callback-commands
    $response->addCommand(new EntityAddCommand($foo, $bar, $baz));

    //'EntityAddCommand' is something I wrote in my own javascript function

    return $response;
  }

  /**
   * Performs a unflagging when called via a route.
   *
   * @param FlagInterface $flag
   *   The flag entity.
   * @param int $entity_id
   *   The flaggable entity ID.
   *
   * @return AjaxResponse|null
   *   The response object, only if successful.
   **/
  public function unflag(FlagInterface $flag, $entity_id) {
    $response = parent::unflag($flag, $entity_id);

    //.... etc, do your own commands for unflagging

    return $response;
  }