sample.html

<!-- default embed code -->
<iframe width="560" height="315" src="https://www.youtube.com/embed/9BHGxXINg-Y" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>

<!-- updated embed code -->
<iframe width="560" height="315" src="https://www.youtube.com/embed/9BHGxXINg-Y?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>

wp-remove-youtube-related.php

<?
add_filter('oembed_dataparse','youtube_remove_related_videos',10,3);

function youtube_remove_related_videos($return, $data, $url) {
    if ($data->provider_name == 'YouTube') {
        $data->html = str_replace('feature=oembed', 'feature=oembed&#038;rel=0', $data->html);
        return $data->html;
    } else return $return;
}
?>