This is a very artificial situation. Two people called Tony who make videos, and you want to specify that you watch a video by one of them, but not specify which one... So realistically this is not a problem.
(2) doesn't work because it states you watched a video of a video (eg someone pointing their camera at the screen showing a video)
(1) means one video made by one of the two people named Tony. Saying "The Tonys" means that you are treating them as a single group of two people. It's all a bit awkward.
So really what is the communication point here?? Why is the name but not the identity of the person who made the video important? What do you mean by "not a specific video?" If you want to say "some video" that is possible, treat "video" as uncountable:
I watched some video that Tony made.
Of course it is ambiguous, but that is exactly the ambiguity you asked for. The fact that there is another person who makes video with the same name is irrelevant but if you insist:
I watched some video that one of the Tonys made.