I once had a question on how to get current multi-site instance name. You can use the accepted answer's code to add another template suggestion based on the name then:
/**
* Implements hook_theme_suggestions_HOOK_alter().
*/
function MYTHEME_theme_suggestions_node_alter(array &$suggestions, array $variables) {
$site_path = \Drupal\Core\DrupalKernel::findSitePath(\Drupal::request());
$site_path = explode('/', $site_path);
$site_name = $site_path[1];
$site_specific_suggestions = [];
foreach ($suggestions as $suggestion) {
$site_specific_suggestions[] = $suggestion . '__site_' . $site_name;
}
$suggestions = array_merge($suggestions, $site_specific_suggestions);
}
Before:
<!-- FILE NAME SUGGESTIONS:
* node--236--full.html.twig
* node--236.html.twig
x node--page--full.html.twig
* node--page.html.twig
* node--full.html.twig
* node.html.twig
-->
After:
<!-- FILE NAME SUGGESTIONS:
* node--236--full--site-default.html.twig
* node--236--site-default.html.twig
* node--page--full--site-default.html.twig
* node--page--site-default.html.twig
* node--full--site-default.html.twig
* node--236--full.html.twig
* node--236.html.twig
x node--page--full.html.twig
* node--page.html.twig
* node--full.html.twig
* node.html.twig
-->
