Are you having a videophone or something. How is nintendo suposed to see your serial if you call them?
So you are probably trying to use the shopping channel which is not working, and now you want to call Nintendo for help?
To my knowledge, the wii contains a small eeprom memory as well, but I don't know what is stored in that. Fact is that it's separate from the nand.
Another thing they can find out of your wii is it's MAC adress. That's unique for every wii as well.
But, I don't know if they use it or not. I do know they are not very helpfull if they find out your wii has been modded.