I believe you were able to play due to early access but that ended at the end of last month.
You are on the wrong website to see your registered FFXIV account. What you are looking at is your game being registered as "This is something I own" on the members site. Not as "This is something I have an account set up for". Sorry if the explination is a bit hard to follow.

Go to this link, log in and click on the mog station banner: https://secure.square-enix.com/account/app
Once you are there, click Account Services and then Add Windows Service. *DO NOT CLICK* add new service account.

This should sort out your issue.