You can give it any serial number, as long as it follow N's algorithm.... as far as I am aware.After installing the downloaded files from NUS, a windows appears which allows you to edit the serial number in setting.txt.
If the serial number isn't the same as the console, It will result in a brick.
Edit: but, if the key's match the console, should it matter if the serial isn't identical?
I've done it to my wii and I have given it PAL, KOR, and JAP regions, and I've also given it the serial number to my other wii without any issues.
But if what you are saying was true then we wouldn't be able to convert a nand to work on anopther wii using betwiin or punetwiin, and I've done that hundreds of times.