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Actually is is not quite true that you are required to install a module more than once to get it to show up in the menu. There are ways and means of getting around each task by using one ore more modules. Actually some content modules will have their own menu for displaying categories as you have suggested. ie Smartsections, wf-channel are a couple I can name straight off my head.
You could also use the menu module by solo, do a search here and you will find it :)
As to the point of duplicating modules, the process is a little more difficult than just renaming the folder. There are many other factors to take into consideration. ie. Database tables, module constants, module paths all within the code of the module.
However, there are many content modules that will allow you to duplicate themselves (again smartsections comes to mind or pico is another one)
Also, look in our FAQ section regarding duplicating modules and what is required.
Catz