In my experience, I have found that not to be the case. I have found that when NW and CO shared a flight, they both had different numbers for all classes, including F. This happend to me about 2 days ago.

I was searching on ITN for my outbound flight. From SEA-EWR it showed F0 for CO, but on the NW flight numbers, F1. I called without hesitation and was confirmed in First. (because it wasn't automatically processed)