I have used non-Enterprise versions of Windows 8 several times to create a Windows on a stick. It works with Windows 8.1 in exactly the same way it did with Windows 8.
You activate Windows when you set it up the first time. If you later plug it into different physical hardware, you have to activate it again (which could become a problem if you use it a lot).
You could use versions of Windows that you don't have to activate, such as evaluation versions or Hyper-V Server.
So while it is technically possible, that doesn't mean the license allows you to do that. Often it says something like: 'you have the right to install this software on a single PC'
I can't say anything about the licencing aspect of this.