In the case of trans folk, the question is whether you're what society sees you as or what you see yourself as—see The Dispute Between Radical Feminism and Transgenderism. If what matters is your idea of yourself, then white men can only be called white men if they identify as white men.
As for Zionists, a game of Oppression Olympics is being played that no one can win. To the Students for Justice in Palestine, a Letter From an Angry Black Woman begins
The student organization Students for Justice in Palestine (SJP) is prominent on many college campuses, preaching a mantra of “Freeing Palestine.” It masquerades as though it were a civil rights group when it is not. Indeed, as an African-American, I am highly insulted that my people’s legacy is being pilfered for such a repugnant agenda.She goes on to claim a number of black civil rights figures were Zionists—but she doesn't quote them, and doesn't note that many people supported Israel in the '50s and '60s before they learned more about the suffering of the Palestinians, and she wisely ignores Malcolm X.
The problem identitarians face with Zionism is simple: What are the identities of Palestinians and Zionists? Both have historical claims to having been oppressed. If you conclude that Zionist is a false identity, are you denying the validity of Israel, a state founded and controlled by Zionists?
Ah, well. Those are my current contenders for the issues that will ultimately cause identitarianism to go the way of Manifest Destiny. But I realize I'm hoping for logic to matter more than it does.