Yeah I'm wondering if that's the issue. Have you tried making another user account on Windows and switching to that? Otherwise maybe clear out %USERPROFILE%\AppData\Local\Ubisoft Game Launcher?What buttons did you have appearing on your Uplay app? Or did you use the site? Safe to assume the have wasn't installed beforehand on the PC? Really wondering if that's my issue... But this is still pretty ridiculous of a problem. "Oh, you share a PC with someone that owns the gave already? Too bad, you're not allowed to have it"
Other thought would be a different PC/phone/tablet entirely, as maybe they store your computer ID or something when you "activate" the free game to stop people from making a bunch of alt accounts and re-selling them.