What does that even mean? Again using this on your FF14 account in no way links it to your Google account at all.
SE generates a unique code that's linked to your FF14 account. You input that code into the authenticator app of your choice. BAM you're done. Google is no where in the loop.
I've literally implemented this system into my own personal web sites and my companies administrative systems.
Here, go read how TOTP works.
https://tools.ietf.org/html/rfc6238
Take a lesson from other good providers. If I want to remember my login on MY OWN COMPUTER it's not any less secure. The one time password is to prevent people from logging into your account from other locations. I'm pretty sure some hacker isn't going to break into my house to use my computer and get into my FF14 account. Even then the OTP is still needed to log into my mog station account.. I'm referring to the game login. Consoles have the option to remember their passwords so they don't have to enter it every time.
P.S. Having to manually enter your password every time on your PC is a bigger security risk than remembering your login. Those that don't have the OTP on their accounts are more likely to get hacked this way if they get a rootkit virus on their PC.