Is is possible to use the new macOS Sierra PiP (Picture-in-Picture) feature in browsers other than Safari?
Chrome 69 now natively supports PiP here.
Picture in Picture for macOS: https://itunes.apple.com/ru/app/picture-in-picture-floating/id1099477261
To be clear, the answer to the OP's question is that no, you cannot use macOS' native PiP mode in any browser other than Safari.
I wish there were some way around that, as I love using the Video Speed Controller Chrome extension to 2x videos. I use this dozens of times a day. But I also love Safari's PiP feature. I can't use both at the same time. :(
The other answers here are providing solid workarounds, giving us the best options available in place of Safari's PiP feature, but just in case anybody's confused, they do not actually utilize the native macOS PiP functionality. They are all different tools trying to imitate Safari's feature.
Further, Chrome's deployment of a PiP feature is not based on the native macOS functionality at all (as of version 71.0.3578.98). It's a custom piece of code Chrome wrote to create the same kind of experience that you get in Safari, but it is severely limited in that it cannot show PiP when you're fullscreen in a different app, which is generally how I use PiP. You have to be in the Chrome window that has that particular tab that you're pulling the video from. You can do this in fullscreen, so long as you stay in that Chrome window. The only way to be working in another app and have the Chrome PiP visible is to have both the Chrome window and the other app's window sitting on the desktop. You cannot go into fullscreen mode.
Sorry – I know that's verbose, but it's kind of a convoluted issue.
Hope that helps someone!
For Chrome and Firefox you need to install a plugin which you can find by search on the store.