In modern Turkish "z" is not the plural suffix anymore, in general.
Oğuz Türks are the southwestern branch of Turkish clan system.
"Gökoğuz" was a group of Oğuz Türks who migrated to northwest and named as Gagauz in modern times.
One of the earliest rulers of the Turkic people is also named "Oghuz Khan".
As a given name "Oğuz" used in memory of and connotes Oghuz Khan.