Switch the render engine to Cycles instead of Eevee.

Designed to be used for rendering in Cycles, the Eevee renderer cannot be used for rendering ID mask output in this way.

But of course, you can use the output from Cycles for use on renders from Eevee. 

The CID is running only on Cycles. But it doesn't matter too much if you use for modeling and final rendering Cycles, Eevee, or Octane. The output produced by CID is mainly for export and use by external software - such as Photoshop.  So if you want to get a Color ID map for Photoshop, just switch to Cycles and make a Render.

Yes, it does. Be sure to always use at least PNG (or EXR) instead of JPG!

Yes, you can, just add a new color ramp stop at the beginning with a different color. (edit10.4.2022: From version 1.2.1 by default)

If you need more colorful contrasts between the color IDS, don't hesitate to add more colors between color ramp stops in the compositor!

Yes, you can add the anti-aliasing node. But for the better selection workings in Photoshop, It is recommended to don't use it and leave it as it is by default. 

If you need just Material ID pass in addition to every render, Cryptomatte is also an option. The problem is that Cryptomatte is designed for use in Compositor. If you export it to PNG - the output is jaggy and not possible to use for clean postproduction. Check the comparing screenshots. 

Yes, it does, check the screen of the brick referce bellow. You get colour with respect alphas, no rectangle polygons here. 

 Alpha transparent mask on textures:

There are many resources on the internet but for example, check this (after 6min30s):

https://www.youtube.com/watch?v=wtdD68irsr8

this one after 1min:50s:

https://www.youtube.com/watch?v=DMPAZbr9op8

or this: https://www.youtube.com/watch?v=MR3ZyZllBrE and many more. 

The CID can be set to a random color generated for all objects with the same material. This procedure is very convenient for post-production in Photoshop. In addition, this output is generated automatically with each render and there is no need to always switch to a different view.