YouTube for iOS and Android feature the picture-in-picture option for quite some time now and it looks like YouTube is aiming to bring the same to the web as well. The company is currently testing out the feature on a small subset of users.
Once the feature is enabled, users can start a video and then click anywhere on YouTube including returning to the homepage, performing a search, or viewing a channel. The video will continue to play in a floating window in the bottom-right corner of the screen. Controls for the minimized window include play/pause, replay, or watch the next clip and the name of the video clip appears underneath, while full playlists can be viewed and scrolled by tapping an icon in the bottom-right corner.
At this moment it is hard to tell if this feature will be default like on YouTube mobile apps or users can disable it if they wish to. There is also a chance that it might be restricted to those with a YouTube Red subscription. Since this is a testing phase, YouTube could still decide not to launch it all together.