You cannot restrict tabs to users. The only thing you could do is link to a Stream channel that some folks don't have access too the individual video's themselves, but that would be very cumbersome but it can be done.
Another option could be not to use stream, and upload the video's to a library inside the SharePoint site connected to the Team, and restrict that to certain users. You could also have a folder inside a files tab with video's and restrict that folder as well.
The "..." allows any user to customize personal Apps and not to control if an App can be added or not to a Team...indeed you can try to select "No" in the "Add to a team or a chat" setting and you will note that this actually does not disable the ability to add Stream as a tab in any team...so to my knowledge, Chris is correct here