Aikodabratt If you bought the game directly from EA, you don't need to register it on the Sims 3 site, as the site should already know you own the game. Just sign in and see whether the site acknowledges your ownership of whatever you bought.
If the site doesn't think you own Sims 3, you'll need to contact EA customer support for help. Say specifically that the Store and Exchange are features of the game and you want access, and the only way is to remove the game from your account and add it back. I would strongly recommend using live chat rather than email. If you don't see the option, try this link:
https://help.ea.com/en/help-contact-us/?product=the-sims-3&topic=missing-game&category=manage-my-account&subcategory=find-missing-game
Make sure you're signed into an EA site (this one is fine) before clicking the link and that the location in the lower-right corner of the browser window is set to U.S. or U.K.
By the way, the product key is in fact the serial code you'd want, but it's already been used because it's already attached to your account. So it won't work a second time.