Programmatically get current multi-site instance name
Thanks to @Clive for telling me about DrupalKernel::findSitePath which takes the current $request to then return the path of the matching multi-site directory.

And thanks to @4k4 for the tip that there's also a service for that: 

/**
 * Implements template_preprocess_html().
 */
function MYTHEME_preprocess_html(&amp;$variables) {

  $site_path = \Drupal::service('site.path'); // e.g.: 'sites/default'    
  $site_path = explode('/', $site_path);
  $site_name = $site_path[1];

  $variables['attributes']['class'][] = 'site-' . $site_name;

}