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I would go with the former, a Google issue, if you want ot troubleshoot it, repair the phone with PCC and try again, it will narrow down the issue. Have you unlocked the bootloader?
Bridge (for Mac)
Alternatives on How to backup Xperias
Ok first under settings/accounts/add account
Then tapping the blue google icon I get the just a sec screen and then couldn't sign in screen
When I tap next I return to the first screen. Incase your wondering nowhere in the process do I get asked for the google account or password
Lets wait for the screenshots to be approved by a moderator so I can see them.