When I was in Afghanistan, one of the interpreters and several Afghan National Police I was with made their own yogurt by straining it and wanted me to try some. To my surprise they gave me a honey flavored yogurt with the consistency of what I knew as Greek yogurt. From that day one I wondered how strained yogurt became known solely as Greek yogurt in the United States.
I recently discovered the answer. No one should be surprised that the answer is in fact: business. In the early 2000s the Greek company FAGE expanded into the United States. FAGE was the first company to massively market strained yogurt in the United States. FAGE gave the yogurt "Greek" in its title, probably because Greek yogurt sounds more appealing than strained yogurt. FAGE became massively successful and now other companies are trying to cash in on what they consider to be Greek yogurt.
The Greek stereotyping of strained yogurt is part of its advertising appeal. How many Greek yogurt ads feature either ancient Greek ruins or images of beautiful Greek islands. The appeal has turned into a stereotype meant to be laughed at as well. However, I challenge anyone to imagine the below commercial with a disapproving secular Syrian Baathist, Afghan Muslim, or Hindu woman instead of an old Greek Orthodox woman. The commercial would be decrying as distasteful and hateful and pulled immediately.