There are several definitions of anonymous.
As per the free dictionary for example, "Having no distinctive character or recognition factor" is anonymous.
Eg: Having an unknown or unacknowledged name: an anonymous author.
As per the wikipedia page on the anonymous group,
"The name Anonymous itself is inspired by the perceived anonymity under which users post images and comments on the Internet. Usage of the term Anonymous in the sense of a shared identity began on imageboards, particularly the /b/ board of 4chan, dedicated to random content. A tag of Anonymous is assigned to visitors who leave comments without identifying the originator of the posted content. Users of imageboards sometimes jokingly acted as if Anonymous was a single individual. The concept of the Anonymous entity advanced in 2004 when an administrator on the 4chan image board activated a "Forced_Anon" protocol that signed all posts as Anonymous"
As per Google,
(of a person) not identified by name; of unknown name.
Through all these definitions the one thing we can establish is the fact that behind every real or fake anonymous persona, there is inherently a very real person who made it. Anonymity therefore lies in the lack of information correlating the real identity and "anonymous identity" of the person. The more the correlation's tangibility, the lesser the obtained anonymity.
For example, voting in a democratic process, votes must be verifiable, but yet anonymous. It's a clear example of how it is possible to have a system where within the system you are anonymous. Votes are termed anonymous because you can not correlate a vote to a specific individual.
There are three limitations to this approach:
1) For anonymity, there can not be a distinguishing factor.
For example, in an "anonymous publishing system" writing works would have distinctive characteristics that could be used for recognition.
Or in a voting system, those close to the person voting would know his vote. Or you can interrogate a person until he gives up his anonymous vote (both of which can be flawed by disinformation).
2) Numerical Strength
Numerical strength makes it harder to tie patterns to one individual.
3) Legal Standpoint
Any system that goes "against the law" will have to prove accountability for the accused illegal actions. If it fails to do so, there will be attempts to shut/disrupt the system's functioning. Additionally as history shows, all its members will be treated equally liable as the anonymous identity that committed the accused crime.
In practice, if you had to authenticate someone, you would need a secret.
If this secret was shared amongst all trusted members based on an identity requirement (how voting is done), then when inside the system with static input their identity would be anonymous. In case of dynamic input systems, the person would have to reduce recognizable marks enough to blend into the system's population to stay anonymous.
The problem in this system that gives you authentication and anonymity, is accountability if data leak prevention fails. That is, if the shared secret is leaked, there is no viable long term defense measure.
If the secret is not a shared secret, but rather unique to a each person, then the system must be designed such that it does not leak session data when making a post. Eg. All posts on /b/ appear as Anonymous. Obviously, the backed should not have logs for IP, etc as well. In this scenario, a leak is isolated to that person's online identity on that system only, thus preserving system security. Since in this implementation there is no correlation between the now selectively isolated user and his real life identity, he is still anonymous.
Obviously, the limitation of this is a reset.
In case of a forum, banning an anonymous member would mean that he can re-register.
In case of a voting systems, banning a compromised account would mean that the user would have to wait until the entire systems' users reset.
Eg: If the counter for voting for a candidate fails, then people can't really complain about it while being anonymous. They would have to wait for next year or complain about it after giving up anonymity.
However, it would work when banning a member from a system that does not accept new members.
So in this case, you can build systems to have authentication and ensure anonymity of actions while within the system, however unless you fake legal documents, being truly anonymous in the system of life is not a possibility.
A 3rd party in this case would be to ensure the system itself is not compromised. But the system is inherently only as safe or as vulnerable as any other software built.
Lastly, BitCoins is another example of where it is possible to have a system with authentication, accountability and anonymity within the system. So yes, it is possible. What's not possible is to guarantee that all successful authentications by the original real identity that formed the online identity.