A somewhat popular 'argument', which notably tends to come mainly from non-anarchists and abusers, goes something like this:

"Of course that guy is an anarchist, because they say they are! It would be hypocritical and un-anarchist to decide that someone can't call themselves that"

This betrays a misunderstanding of what anarchism is about, and it follows the logic of hierarchy, not that of anarchism. When they make this argument, what they're *really* saying is:

"You don't have an appointed authority, therefore everyone is free to do whatever they want, because nobody is allowed to tell them off!"

That's not what anarchism is about. "Not having authorities or rules" isn't the point. Anarchism is about eliminating *hierarchy*; about socially organizing and fostering healthy communities through collaborative and participatory means, rather than through 'authority'.

If someone does not live or speak by these principles, then you, as a part of that anarchist community, are entirely within your rights to say "they are not really an anarchist". This is no different from calling out any other problematic behaviour in your communities.

Or to put it differently: you saying that they are not truly an anarchist is not asserting that you have any sort of special authority to make that decision; it's asserting that you and everybody else carry a *shared responsibility* to call out misuse of the term, because that misuse harms others.

And by the logic of how anarchism works, it is everybody's job to do so, as long as it happens in good faith.