Greek here. Greek yoghurt is not strained. The "Greek yoghurt" sold outside Greece, is. In Greece unstrained yoghurt from sheep or cow milk is sold in supermarkets (and everywhere else) alongside the strained cow milk yoghurt people outside Greece are familiar with and know as the only "Greek yoghurt". To Greeks, that "Greek yoghurt" is "bag yoghurt" (because it's strained inside a cheesecloth bag).

Btw, that "Greek yoghurt" is, 90% of the time, Total by Fage (Total is the name of the product, Fage the brand), including the ones that come together with a pot of jam on the side, as immortalised by Silicon Valley's Erlich Bachman:

https://www.youtube.com/watch?v=ZR_taax1TUc

Unless it's Greek style yoghurt which is invariably exactly as wrong as it sounds.