A loooooooong time ago, Nintendo made it impossible to change your Wii Shop Channel region. How does Nintendo identify your console?

It's not impossible, it's just impossible to use one shop account in two different regions. You can still switch if you:
- Delete your current shop account
- Run AnyRegion Changer and switch the Shop Channel region
- Then you can start the Shop Channel and make a new account with the other region.
So basically they give you an account which is linked to your region and switching that region with the same account results in an error code.

Thanks.