Extra url paths as variable

The correct way to register the rewrite rule and rewrite tag for your case is:

function custom_rewrite_rules() {
    add_rewrite_tag('%nights%', '([^&]+)');
    add_rewrite_tag('%people%', '([^&]+)');
    add_rewrite_rule('house/(.+)/(.+)/(.+)/?$', 'index.php?house=$matches[1]&nights=$matches[2]&people=$matches[3]', 'top');
}
add_action('init', 'custom_rewrite_tag', 10, 0);

The code has been tested and works correctly.