Yes, the white serial number is stamped into the rim alongside the (wheel coloured) pat number.
The serial number is unique per wheel (or per pair, which is what I was wondering)
The pat number is the same on all wheels as far as I've seen - the exception being earlier so-called 'metal hub' wheels which used a different (but still in common) pat number.
There only seems to be the one generation of early alloy hub wheels that had the serial number. I thought it might be possible to tell from this number whether two wheels belonged to an original pair. Though as it's impossible to tell with other generations of wheel there's little to be gained from the exercise.
The following pic shows the pat and serial numbers. Should the white serial number be exactly the same on the matching wheel in the pair? Does anyone care?