My question is simple, if I have a subscription to a website that has all my identifying information and in return they give me a video to watch, do they ever put my information inside that video? Is it even possible to encode a mp4 file with an email address or name inside it? If that's even possible what programs would I use to look in the files to see if any information is in there? Please help!
Sure it is possible.
If they store the information as metadata, you should be able to see it with any software that can view metadata, as long as they store it in common tags. If they insert it as plain text in uncommon places, you'll need to open it raw and search for the strings. If the text is encoded, you may never find it.
That said, it is trivial to remux the content and strip anything that may be there.
If they put purchaser info in a watermark or something like that, it becomes more difficult to remove, but not impossible. That is highly unlikely though as that would mean they are creating unique content for every purchaser, and that is resource intensive.