Indeed, anonymity disappears if you authenticate... using a VPN is like wearing a mask; people don't see your face and cannot guess your true identity. But if, wearing a mask, you shout your own name and proceed to demonstrate that you are really the person you claim, then your anonymity no longer exists.
When you use your VPN, you make it so that outsiders cannot see "who" you are beyond "someone who uses that specific VPN server presently". If, under this context, you authenticate on Facebook as "Bob the Supreme Warlord" then outsiders can say: well, there seems to be at least one user of that specific VPN server who is also the individual known by Facebook as "Bob the Supreme Warlord" (maybe Facebook knows more about the true identity of Bob). If Bob the Supreme Warlord used Facebook to publish photos of himself doing war supremely, then now the outsiders know Bob's face.
An interesting point, though, is that without the collaboration of the VPN server maintainer, outsiders can only see that "one user of the VPN server downloads this and that" and "one user of the VPN server connects to Facebook as Bob the Supreme Warlord", but they don't really know whether this is the same user -- but they can make guesses. For instance, if someone repeatedly uses the VPN server to engage into intellectual property theft, and another user repeatedly uses the VPN to connect to Facebook as Bob the Supreme Warlord, and these two seem to always be active at the same time, then outsiders (say, police forces) may use their brain cells and infer that Bob the Supreme Warlord might be Bob the Supreme Movie Downloader.
And, of course, any anonymity offered by a VPN server tends to evaporate like morning dew when the FBI knocks on the VPN sysadmin's door at 06:00 AM, and courteously but firmly suggests that flourishing connection logs might be a capital idea at that juncture.