Is my web3name automatically linked to my KILT address?

No, your web3name is connected to the decentralized identifier (DID) you generated when creating your Sporran account. Linking the name and DID address to any of your addresses requires an extra step – see the instructions in the guide here.  

NB - don’t link addresses that you don’t want connected to your name! If you want to keep an address private, don’t link it.