Customizing subject in comment notification e-mails

If I understand you correctly you want to change the subject line of the email. The hook of your code only changes the content of the e-mail. There’s a different hook for the subject:

add_filter('comment_notification_subject', 'wpse_228315_comment_notification_subject');

function wpse_228315_comment_notification_subject($subject, $comment_id){
    return "New comment";
    }