The topic you refer to is wildly inaccurate. These statements are utterly false:
Currently there is no support for custom menu variables.
So the bottom line is: no custom properties in menu definitions.
The params menu entry property is described here:
https://gohugo.io/content-management/menus/#properties-front-matter
Examples of menu entries with a params table:
https://gohugo.io/content-management/menus/#example-front-matter
https://gohugo.io/content-management/menus/#example-site-configuration
The .Params menu entry variable is described here:
https://gohugo.io/variables/menus/#variables
An example of rendering a menu entry parameter:
https://gohugo.io/templates/menu-templates/#menu-entry-parameters
I have no idea why you placed menu items under the site configuration’s params table, but this is the wrong way to do it.