No, the “non-fungibility” simply means that anyone who creates an NFT with the same link will be distinct from your link to the image, even if the actual URL is the same. Both NFTs can also be traced back to when they were created/minted because they’re on a blockchain, a property called provenance. If the authentic tokens came from a well known minting, you can establish that your token is “authentic” and the copy token is a recreation, even if the actual link (or other content) is completely identical.
Nothing about having the “authentic” token would give you actual legal rights though.