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It is not illegal to join Anonymous because you cannot join. Officially, there is nothing to join, although the collective does provide instructions on joining.
Confused? That's because you're supposed to be. Anonymous is a collective of computer coders, hackers, protesters, and geeks who are loosely linked on the web and are, well, anonymous. They do not use their real names. But Anonymous does instruct on affiliation, and following these instructions is not illegal per se.
Crime Is Illegal, Not the Collective
Affiliation with the organization is not illegal (or public). But crime is, of course, and hacking is a crime, and Anonymous members are, famously, hackers. As such, where your Anonymous "membership" gets problematic is if you commit a crime in the name of the collective of "masked avengers."
Showing off about your Anonymous exploits on or offline is also considered very poor form in the community and it puts everyone else at risk. Of course, if you leak your name because you want fame for your technological exploits, you also personally risk arrest and charges ... Plus, no respect.
Forget Your Identity
Anons -- the name for individuals associated with the group -- are not supposed to seek personal attention. They take refuge in anonymity and this is meant to protect people who do engage in illegal activities, like hacking.
The Anonymous "How to Join" section provides as follows:
Hackers Fight Terror
In the wake of the terrorist attacks in Paris last week, Anonymous declared "war" on the Islamic State or ISIS, the group claiming responsibility. Now the techies are taking on the terrorists by exposing their Twitter accounts and even, reportedly, the physical address of an ISIS recruiter.
But according to the Independent, Anonymous has gotten it wrong before, outing alleged terrorist recruiters who were not ultimately what hackers thought. That is not only potentially criminal but also great fodder for a civil suit if the hacker's identity is uncovered.
The long and short of the story is, don't get involved with Anonymous if you want to ensure that all your activities are legal and that everyone you associate with operates within the law. In the words of the collective itself, "You might want to consider joining a publicly and legaly [sic] known activist movement where you don't need to live like Batman."