I was given a game I already own.

You did not win a game you already own, you were given a game you already own. The difference being that you did nothing against the rules in the former case. You should appeal to the GA creator to get the correct game as stated in the rules, or you could allow the creator to delete the giveaway if they are unable to gift you the correct and stated game. Doing otherwise would lead others to believe you broke the rules by not activating a game on your own profile. This was most likely a minor error on the creator's part, thinking he was just giving you additional DLC, but it's a different ID instead.

Ask the giveaway creator to provide the correct key. If he/she does not do so within a week, mark as "not received".

If the user is new around here, you might agree to them deleting the giveaway.

yeah that one is kinda a understandable mistake.
one would think the gold edition includes the same base game, but in this case it doesn't.
personally i'd give permission to delete on that one, but that is your decision to make. you can either delete it, or try and make them provide the listed games key. if they won't comply if you decide to pursue the game route, then all you can do is mark not received after the 7day period.

Thank you for the answers. I'll try to contact the giveaway creator and let him or her decide whether to delete the giveaway or give me the correct key. But in this case I don't mind if it's just deleted. Again thank you.

if he doesn't give you the correct game, it could be taken as a fake giveaway. depends on how you feel about reporting him.

I contacted the giveaway creator and we agreed to just delete the giveaway. It seems to have been some kind of misunderstanding. It still shows that I've won a giveaway, but I guess it will stay that way until support deletes the giveaway.