I don't think that the Chrome developer tools are Material based, it doesn't appear so. Also, the post you mentioned says that they're using CDK for the console, not the Material components. CDK is a library for shared common functionality (I didn't use it - from what I know) and that might be used there.
No idea on draggable cards, sorry... . From the top of my head I doubt if that's possible. I didn't see on Google Material Design documentation such an option. They are expand-ables but perhaps not draggable components. If they are, maybe just 'plant' a card on such a 'floating deck'?
Good luck,
Boaz.