NFTs neither do nor can enforce uniqueness too. Blockchain is a method to, as you put it, trustlessly couple a particular instance of a datum with a keypair.
Game developers can duplicate the blob describing in-game asset in their own database or on the blockchain. For off-chain data blockchains are nothing more than a datastore which provides immutable link between a datum and a keypair.
So yes there is an original keypair that establishes the data. But ones established it can be fixed in a verifiable way. A game client, community of art lovers, government agency, social consensus, then recognizes that nft as corresponding to some offchain asset. It cannot be later changed, duplicated, etc without some social consensus to do so. You don't need to trust a third party with control.
