The lawsuits are mainly in the U.S., aren't they? As far as I know, in the EU the situation isn't really disputed, since origin labeling is stricter. Here in Denmark there are three common categories of labeling for this style of yogurt: 1) "Greek yogurt", which implies it's imported from Greece; 2) "Turkish yogurt", which implies it's imported from Turkey; and 3) "strained" or "Greek-style" or "Turkish-style" yogurt, which doesn't imply a geographic origin.
Fage oddly has nearly zero market share in Denmark though, despite Greek yogurt having a good market share. The dominant brand is the Danish brand "Nostimo", part of the Danton Trading group run by a Greek-Danish entrepreneur. The yogurt is all sourced from Greece, but the operation is run mainly out of Aarhus.