I am trying to set up a number of different content types and have come across a road block. I am trying to create a "Policy Document" content type and a "Training Document" content type. Within each of these I want a column called "Category is", with the input type "Choice". However in each of these content types I need the choices to be different. For example within "Policy Document" I want Internet, Vehicle, Expenses; and within "Training Document", I want Word, Excel, and Powerpoint. Is this possible?
You need to create your columns and your content type declaratively (through xml in a feature) or programatically (console app, event receiver). Like @thantos said both columns can have the same Display name, yet the Internal Name, the Name and the GUID need to be different
Just make the column on the content type, but with the same name, that should work.
It is not recommended to have 2 columns with the same name in Site Columns.
If you need the field to be the same, but the selection options to be different, you merely reference to the column in your second contenttype using it's GUID and specify the new choices available.