What I've read just now is what I'd deliberately not gone looking for up to this point in case I found this out.
Linux and Windows happen to require that the executables that access that information be run with root privileges, but it looks like that's merely an affectation.
It seems that any old piece of software, without root or other privileges, can independently run the CPUID instruction that obtains a processor's serial number.
I do not like this one bit.