I once had a question on [how to get current multi-site instance name][1]. You can use the accepted answer's code to add another template suggestion based on the name then:

```php
/**
 * Implements hook_theme_suggestions_HOOK_alter().
 */
function MYTHEME_theme_suggestions_node_alter(array &$suggestions, array $variables) {
 $site_path = \Drupal::service('site.path'); // e.g.: 'sites/default'
 $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:

```html
<!-- 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:

```html
<!-- 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
-->
```

 [1]: https://drupal.stackexchange.com/q/257325/15055