The site claimed that I have a game that I don't have

Exactly. It happens with every game. If they are free for a week or weekend, they disappear from here, and free games automatically get linked to your account.
Try it again once the free period is over, which I think is :D

Ahh so that's what it is. I wasn't online during the weekend so I didn't know this.

Do not sync on weekdays. Automatic sync will do it again on weekdays.
EDIT: After reading properly. Well, I don't know if you can get banned that way :)

If you were previously entered, you're Ok. You won't be removed from the GA. And the person giving the game will see that you don't have the game when they check. (At worst, they'll have to wait after the weekend to confirm this.)

The only issue is for those who synced during the weekend can't enter any GA until the game doesn't show on their account anymore. If you actually play the game while it's free, it stays on your account for 2 weeks.