I found no clues in the developer documentation, but equalify claims that it integrates into the desktop application. They don't offer a free version so I couldn't try it out.
Is it possible/supported to create extensions for the desktop client and is there some hidden documentation available?
I'd be interested in developing one, but so far I had no luck in finding any documentation.
It used to be possible using the Spotify Apps API. I think you can still manually sideload apps, but there is no App Store or similar. Moreover, support for old Spotify Apps is about to be dropped completely in the future.
So no, there's no way to extend Spotify as far as I know.