I solved this issue on an Xperia Z1. I was unable to connect to Facebook using the Sony Albums app, or get a connection to facebook using the "Xperia with Facebook" app, all it would do is loop back to the connect screen, give me the license terms, then loop back to the connect screen.
The fix for me was a little odd, but it's 100% working now. In the Sync Settings I allowed my Facebook Contacts to Sync (I don't normally do this as it bloats my contacts to much), once my facebook contacts had synced, I installed the latest Facebook messenger app from the play store. I then went back to the Accounts settings, added an account for "Xperia with Facebook", and boom, connected instantly, and now works in my Albums app to.
If I only enabled the Contacts sync, it still would not work, so somehow, a cobination of syncing contacts and downloading the Facebook Messenger app worked. I have since disabled Facebook contaxt sync, and "Xperia with Facebook" is still working.
Strangely I could not get it to connect with "Xperia with Twitter" either, but just installing the latest Twitter App from the Play store fixed that issue.
I hope this helps.