I searched Vielfliegertreff and FlyerTalk and there are many reports of upgrades to LH flights under LH numbers ticketed on UA stock.
So it is possible and you being offered this option is not an error (the LH website doesn't list the ticket stock as a requirement for cash upgrades either).
The ticket definitely has to be reissued and that's the problem.
Theoretically the ticket could be transferred from UA to LH for reissuance on 220 stock but I doubt they'll do it.
If left untouched you'll definitely run into issues since the reservation and ticket are out of sync.

I wouldn't trust LH when they say "everything looks good".
Have you tried looking up the ticket on saudia.com using hte ticket number?
I don't know if it would work with a 016 ticket that has LH flights on it but it might (it does work for my LH reservations and gives you all the detail on the actual ticket).