The problem with this is that it makes the word ultimately meaningless: if "geek" only means “the set of people who say that they're geeks," then "geek" doesn't mean anything at all. You might as well say "fleegash means people who say that they're fleegash."
This is a pretty good refutation of what I'd call the "ontological" understanding of self-identification. In this view, saying that you're an X is what causes or constitutes your X-ness. You make yourself an X purely by identifying as such. I think that for some identities, self-identification may be a necessary component of having that identity. For example, I don't think someone can really be a Christian if they don't accept that label for themselves (though their words and actions may still be compatible with, and helpful to, Christianity). On the other hand, self-identification is not always required -- for example, we can insist that a person is cis even if they don't know the term or even if they petulantly refuse to accept it. But making self-identification the sole component of an identity leads to the sort of meaning-destroying absurdity of Ozy's "fleegash."
Ozy goes on to defend what I'd call an "epistemological" understanding of self-identification. In this view, the self-identifier has better knowledge of their own identity than anyone else. Though they could be mistaken about their identity, nobody else is really in a position to know any better, and thus we should defer to the person's own self-identification. The problem with this is that in many cases an outsider does have access to good information. I could go around self-identifying as a natural-born Canadian citizen all I want, and I may believe it with my whole heart -- but someone else could order a copy of my birth certificate from the state of Ohio and disprove my claims. Moreover, the epistemological understanding doesn't account for disingenuous self-identification. Maybe I know full well that all my ancestors were from Sweden, but I've decided to self-identify as Native American so that I can go to a sweat lodge. If I'm not making honest use of my special self-knowledge, it doesn't carry any weight.
I think there's a lot of value to the epistemological understanding of self-identification, but I also think it doesn't give a full picture. I would add the importance of a "methodological" understanding of self-identification. In this view, it produces the most socially beneficial outcomes to defer to sincere self-identification. It's a matter of adding up the costs and benefits of deference to self-identification versus policing of identities. Deference will make sure to include everyone who legitimately has the identity in question -- but risks allowing people who are mistaken or disingenuous to claim that identity too. Policing excludes people who shouldn't actually have the identity in question -- but it also excludes some of those who ought to be able to claim the identity, both by directly incorrectly denying them membership, and by exerting a chilling effect on people who see the policing going on and don't want to expose themselves to it even if they would ultimately pass the test. These costs of policing fall particularly heavy on people who are new and just exploring the possibility of accepting a certain identity, and on people who have intersectional marginalizations and thus don't fit the policers' paradigm of a person with that identity. Weighing up these costs and benefits, it is usually (but not always) more harmful to engage in identity policing than to defer to self-identification. This is particularly true for public applications of policing, such as by organizers of events with restrictive attendance rules. A trusted friend may be able to take me aside and say "dude, I know you're not gay, so stop saying you are," but the people running a gay-only health workshop should err on the side of trusting anyone who shows up saying they're gay.