I've found a way to enable this menu in Chrome: just right click the video twice in the same place. The first right click will trigger YouTube's contextual menu, while the second right click will display browser's menu. Features like "save video as", "copy video URL" don't work, but you can use this trick to watch videos on repeat. Just click "loop" and you can watch your favorite YouTube video over and over again. To deactivate repeat, right click the video twice and click "loop" once again.
Obviously, this only works for YouTube's HTML5 player, but it's enabled by default in Chrome and most YouTube videos use it. It doesn't work for embedded videos.
There are other ways to loop YouTube videos: from adding videos to playlists to using the "loop" parameter, installing extensions or using sites like Infinite Looper.
To enable the menu in Firefox, you can Shift click the YouTube player. There's no loop option, but you can find features like "save snapshot as", "view video" and more.