As Apple has never sold a mouse with a middle button, macOS has never had native support for assigning any custom actions to that input.
However, there are a variety of third-party utilities which will enable this, such as BetterTouchTool or Multitouch.
As of Catalina (MacOS 10.15.5) and Big Sur (MacOS 11.2), I can verify that this is now the location of the desktop background images.
If you want to navigate to it, you can press command K and paste the text below in.
1.4 GB too. : [
You'll need to turn off System Integrity if you wish ...
Option 1: Mission-Control Setting "Displays have separate Spaces"
I'm not entirely sure if this is what you're looking for:
If you go to "System Preferences" ➝ "Mission Control", then uncheck "Displays have separate Spaces":
This will couple the spaces of all screens. So if you swipe or switch to another space on one ...
If your 'pairs' are always the same combinations 1&6, 2&7 etc, then switching off 'Displays have separate Spaces' will solve it - though you'll have to set your pairs up again initially as it won't combine your existing screens intelligently.
If you need anything more complex/flexible, then you'd probably be looking at something 3rd party.
I use the "3-finger-pushup" method for this;
Make sure you have "Mission Control" enabled
Execute the "3-finger-pushup" move; the menu bar is replaced with a list of your desktops:
Select one of the app windows & drag it to any desktop you like.