Hmm. Two thoughts in response to the original question:
1.) If you "need" the State to constitute yourself as an anarchist, you're not a very good anarchist ("good" in the sense of "real" or "effective," not as in "Are you a good witch or a bad witch?"). More than that: if you need _any_ kind of oppression or domination to be an anarchist, then you're playing a losing game. Anarchism that is thought and practiced as something that is solely or even primarily about _negating_ something else is poorly developed. Fortunately, we've got more to say and do than that. Here's my translation of an entry from Daniel Colson's _Petit lexique philosophique de l'anarchisme de Proudhon à Deleuze_:
"_Positive anarchy_ (see _negation_, _more than oneself_, _equilibrium of forces_, _tension_ and _power of the outside_). A Proudhonian concept that also serves to dissociate the libertarian project from the negative and from the _ressentiment_ that the struggle against relations of domination is always likely to cause when it is not transformed at once from _refusal_, _rupture_, and revolt into an affirmative force able to recompose the world differently, in an emancipatory fashion. Positive anarchy is the affirmation of a dynamics and a new arrangement able to liberate collective forces from their confinement and enable them _to do all that they are capable of_."
(The entry on "emancipation"/"affirmation" goes into greater detail about this.)
2.) On the other hand, there might be a limited truth to the notion that the State is what constitutes anarchists as anarchists. One way of describing a world in which we have "won," a world in a condition of positive anarchy, is to say that "everybody is an anarchist"; another way of describing the same world is to say that "nobody is an anarchist," because without the forces to which anarchism (the historically specific movement and body of ideas, not the same thing as "anarchy," which is routinely practiced by people who have no contact with it, and which is as old as the species, as far as we can tell) was opposed, "anarchist" no longer has any meaning as a particular identity. The goal of anarchists might be, from this perspective, to abolish themselves as anarchists.
This auto-destruction might be what Andrew X was getting at in his widely-read 1999 essay, "Give Up Activism," where he suggests that the "activist," as "a specialist or an expert in social change," needs to be abolished (see http://www.vgpolitics.f9.co.uk/00913.htm
). What makes this practically interesting, instead of just abstractly paradoxical or a recipe for inaction, is that folks like the Umsonst groups translated this into experimental practices aimed at turning ordinary, unengaged passerby into the "activists" (see Anja Kanngieser's essay, "Gestures of Everyday Resistance," at http://eipcp.net/transversal/0307/kanngieser/en