Till now to see a video in Picture-in-Picture mode in Chrome or Chromium browser such as Edge, you need to right-click twice on that video on YouTube followed by selecting the PiP option, you can avoid doing this in new Edge browser by adding PiP control to Global Media Controls UI.

Google has developed Media hub or Global Media Controls UI for Chrome and enabled it by default in the stable version, this feature lets you control audio or video playing in the browser. Like Picture in Picture, you can also make media controls work in the new Microsoft Edge browser.
Recently Chrome 82 Canary has got Picture in Picture icon in Media Controls behind a flag, now the latest Chromium update has upstreamed the same “Global Media Controls Picture in Picture” flag to Edge 82 Canary build, which you can now enable to display Picture in Picture control in Global Media Controls UI in new Microsoft Edge browser, here is how that can be done.
1. Launch Microsoft Edge Canary 82.0.442.0 or later
2. Visit edge://flags page and turn on the following two flags and restart the browser.
Global Media Controls
Global Media Controls Picture in Picture

3. After relaunching Edge, visit YouTube and play any video, you’ll notice the GMC icon on the toolbar, which when clicked, shows PiP icon along with other controls to control music, audio, video, and more media in new Edge browser. Click on the PiP icon to enter Picture in Picture mode for that video, you can get back to the origin tab by clicking on the back to tab button in Picture in Picture window.
Related articles:
Chrome Picture in Picture mode now shows Previous and Next Track buttons for Playlists on YouTube
Chrome on Android is getting Picture in Picture mode
Google Chrome updates Picture in Picture UI
You can now mute videos in Picture in Picture Mode in Chrome and Chromium Edge