What is the term for arbitrary disqualification?

A term that comes to mind is petty or - taken in another direction - discriminatory, but perhaps you are looking for the "No True Scotsman" or "Appeal to Purity" fallacy

No true Scotsman, or appeal to purity, is an informal fallacy in which one attempts to protect a universal generalization from counterexamples by changing the definition in an ad hoc fashion to exclude the counterexample

Person A: "No Scotsman puts sugar on his porridge."

Person B: "But my uncle Angus is a Scotsman and he puts sugar on his porridge."

Person A: "But no true Scotsman puts sugar on his porridge."