Adding Additional Attributes in Script Tag for 3rd party JS

you can try to use the script_loader_src filter hook e.g:

add_filter('script_loader_src','add_id_to_script',10,2);
function add_id_to_script($src, $handle){
    if ($handle != 'dropbox.js') 
            return $src;
    return $src."' id='dropboxjs' data-app-key='MY_APP_KEY";
}

Update

i just figured it out myself that the src is escaped by esc_url, so looking a bit more i found the clean_url filter which you can use to return the value with your id and app key data :

add_filter('clean_url','unclean_url',10,3);
function unclean_url( $good_protocol_url, $original_url, $_context){
    if (false !== strpos($original_url, 'data-app-key')){
      remove_filter('clean_url','unclean_url',10,3);
      $url_parts = parse_url($good_protocol_url);
      return $url_parts['scheme'] . '://' . $url_parts['host'] . $url_parts['path'] . "' id='dropboxjs' data-app-key='MY_APP_KEY";
    }
    return $good_protocol_url;
}