Can you add other accounts with much success?
If you have another Google Account - or a friend is willing to help - sign into that account and see if the same thing is happening. That might shed some light as to whether it's an iPhone problem or a Google Account problem unique to their system or your account.
In the case it's just your account, I'd contact Google's support as they may be able to flick some switch to make it work.
IF the iPhone is allowing other accounts and Google Accounts then I'd doubt it or iOS is the cause of the issue.
Definitely an odd one, let's get it fixed so we can find a solution to share if it happens to others.