The term chicken and egg is not absolute because every term that exists will mean something slightly different to different people. However, simply because a term isn’t 100% agreed upon doesn’t render that term unmeaningful and false. There is no exact definition of what jeans are, for example, and what qualifies as jeans, but most people can still agree that jeans were invented by Jacob Davis and Levi Strauss in 1873.
Furthermore, even if there have been many types of chicken-like creatures, that doesn’t mean that the origins of the modern chicken, or the gallus domesticus, is completely unmeasurable. We have modern samples of their eggs, their genetic traits, reproduction cycles, and all other information needed to investigate their origin. Most origin tests on this issue have also been done on the gallus domesticus. Maybe there comes a point where we trace chickens so far back that we simply conflate them with dinosaurs that have developed tiny wings, but that doesn’t mean that this question is an absolute false dichotomy and can’t even be answered for the most basic farmhouse chicken.