I wrote a user script that does that!
Available on my GitHub, on Greasy Fork and OpenUserJS, under the MIT license.
What it does
It checks if it's on a video page (URL contains /videos/) and if so:
It checks every half a second if the popup appeared, and if so - removes it.
It also cancels the next video from playing automatically.
The code snippet
If the video is made available to a custom list only, then the ability to download and/or share are removed from the basic interface.
That being said, there is surely a way for someone who has access to the video to hack and get a copy of it and then redistribute it on their own, but it is not done easily or through the normal usage of the facebook ...
After searching the web for few hours, apparently there is no current way to disable that feature.
There are some question threads on Facebook's help community that are a few months old, as like the one in the link and none of them got answered by Facebook yet.
How can I turn off the "up next" video feature?
I don't think this is a new feature unless I'm missing an announcement somewhere.
You can just use https://www.facebook.com/search/USERIDHERE/videos-uploaded
or in search phrase format "Videos uploaded by NameofUser"
"Videos uploaded by Mark Zuckerberg"