Drupal - HTML mail without using modules

Without any module it is not possible to send HTML emails. This is because the default plugin the MailManager class (which is a plugin manager) uses to effectively send emails contains the following code. (See PhpMail.php.)

public function format(array $message) {

    // Join the body array into one string.
    $message['body'] = implode("\n\n", $message['body']);

    // Convert any HTML to plain-text.
    $message['body'] = MailFormatHelper::htmlToText($message['body']);

    // Wrap the mail body for sending.
    $message['body'] = MailFormatHelper::wrapMail($message['body']);
    return $message;
  }

The comment clearly states: Convert any HTML to plain-text.

If you look at MailFormatHelper::htmlToText(), you will see the code that is converting the HTML markup in plain text, in particular the following that changes any <a> tag as you saw in your test email.

  // Replace inline <a> tags with the text of link and a footnote.
  // 'See <a href="https://www.drupal.org">the Drupal site</a>' becomes
  // 'See the Drupal site [1]' with the URL included as a footnote.
  static::htmlToMailUrls(NULL, TRUE);
  $pattern = '@(<a[^>]+?href="([^"]*)"[^>]*?>(.+?)</a>)@i';
  $string = preg_replace_callback($pattern, 'static::htmlToMailUrls', $string);
  $urls = static::htmlToMailUrls();
  $footnotes = '';
  if (count($urls)) {
    $footnotes .= "\n";
    for ($i = 0, $max = count($urls); $i < $max; $i++) {
      $footnotes .= '[' . ($i + 1) . '] ' . $urls[$i] . "\n";
    }
  }

You need a module that defines a new mail plugin using a different implementation of the format() method and make the MailManager class use it. As said in the documentation for MailManager::getInstance(), the class uses the configuration in system.mail.interface. (The array returned from \Drupal::config('system.mail.interface').) In your example, an array value for custom_forward_formard (e.g. 'custom_forward_forward => 'custom_forward_mail', where custom_forward_mail is the plugin ID for the class implemented by your module) would do the trick.