I have the “show related videos” box unchecked. I don’t see them on my laptop and phone, but my client is seeing them on two computers and iPhone. He has the current browser for Chrome, is there any issue with older operating systems?
I hadn’t noticed it was a checkbox and had tried to add ?rel=0 to the end and that actually didn’t work. But when I discovered the checkbox and removed it, I’m totally stumped why it works on mine but not on another.
We reported that YouTube had some issues with embed parameters around the time of your posting this request. This problem affected a vast number of WP and non-WP sites. Fortunately, they were able to solve it within a day. Perhaps your issue was part of this fix?