As for your yogurt example, perhaps it's because it's less important to those of us who live really far from Greece. It might not even be desirable to have yogurt that was shipped halfway around the world!
Is it yogurt made from Greek people? Is it yogurt made by a Greek company located in Spain? Yogurt made from Greek dairy, outside of Greece? Yogurt made in Greece from Italian milk? Yogurt made in Greece from Greek milk, but in the Australian style? I can think of all kinds of reasons folks half a world away would care much more about what style the product is, than would care about where, precisely, it was produced.
Of course, when you're in a trade union with a country that has a vested interest in having a monopoly on a product, obviously they're going to be pushing semantics in order to come out on top.
For those of us not so close to Greece, having to call it Greek-style yogurt is just more word soup. (also, this happens all the time in the US; the 'Greek' would be written really large and then 'style' in much smaller letters)
OTOH, it's pretty clear-cut in the case of a product that claims to contain a natural ingredient, but have none of it, and not even have any semblance of the benefits of the actual product.