Assigning the middle mouse button like this will affect applications, like AutoDesk Fusion 360 (personal use version is free!), where the middle mouse button has a real function (panning for example).

Because of this trick, I had this issue where the panning with the middle mouse button on a Mac did not work with Fusion 360.
Note that any double click trick, like this one, potentially will come with adverse effects in other applications.

In my opinion, assigning double click to the middle mouse button should not be that hard.
After some tinker, I did find a solution that works on your Mac, without any extra software.

This is how:

1) Go to System Preferences - Accessibility - Pointer Control - Alternate Control Methods (don't ask how I got there 🤣 ).

2) Check alternate pointer actions and click the "Options" button.

3) Select the "Double Click" row, and double click the "Unassigned" phrase on the left (can be something else if you already had this assigned to something).

4) Select Physical Switch (you'll see Facial Expression as an option incase you have webcam), click "Next" and then press the middle mouse button (or mouse wheel) to register this as the button that should trigger a double click.

You should see something like "Mouse Button 2 ..." in light grey printed in the "Name:" field.

Click "Done" - you can already test the button - and "OK" to close these settings.

p.s. use "Reassign" if that went side ways and/or you'd like to assign a different button.