Hi!
The same problem occured to me as well. I contacted the square enix support, but their suggestions didn't help. I managed to find a temporary solution, which isn't the greatest, but at least works.
Here's what helped me:
Go to settings on your PS4 --> Application Saved Data Management --> Saved Data in System Storage --> Delete, then chose Final Fantasy XIV and delete the settings for the system, not for the characters!
You should be able to log in then, but you have to repeat it, every time you want to log in.
If I find a permanent solution, I'll let you know.