I found the solution:

In the constructor I had to add this

public function __construct()
{
 // your code goes here
 
 add_action( 'plugins_loaded', [$this, 'load_text_domain'] );
 
 // your code goes here
}

public function load_text_domain() : void 
{
  load_plugin_textdomain('your-plugin-textdomain');
}

This finally made sure that the language displayed followed the settings of the user profile and not that of the website language settings. Of course you have to write your own classes constructor. Just make sure it adds the action ‘plugins_loaded’ with the call to the public function that loads the plugin text domain and you are good to go.