At least for non-yogurt products, I suspect that the term “Greek yogurt” is used in package labeling just for advertisement purpose.
In the UK, “Greek yogurt” needs to be produced in Greece in order to be labeled as such. For this reason, there are lots of “Greek style” yogurt (they may contain additives to create the similar texture).
In the US, there isn’t a such regulation, but there is a lawsuit filed in 2014 against a major yogurt producing company in respect to its use of the term “Greek yogurt” (and the term “evaporated cane juice” and its “0% claim”). I wonder if there will be a similar regulation (as in the UK) at some point in the future…
According to The Washington Post, the Greek yogurt trend seems to be fading, and “yogurt drinks” are now in fashion.
“Greek yogurt is no longer the trendiest yogurt” by The Washington Post (2016/10/13)