I don't recall the client ever asking for a registration code. IIRC you register it on the mogstation website. Once it's connected to your account you should only need your user ID and Password (and one-time password if you do that). Are you using the current 64 bit installer? Maybe delete whatever was installed and the installer and download a fresh one.

https://na.finalfantasyxiv.com/lodes...2ba7addc06a99a