Indeed it was used in all formal Byzantine empire -- later "Ottoman".
But it was known far beyond that: "The cuisine of ancient Greece included a dairy product known as oxygala (οξύγαλα) which is believed to have been a form of yogurt" (Wikipedia), and even further to pre-history (5000 BC)
Modern Greek has seen a great deal of lexical replacement. It uses new words for things like "bread" and "water" that are different from what were used in the Classical era, even though bread and water have never changed. There are even cases where Ottoman vocabulary has replaced native Greek words. It is not inconceivable that the word for oxygala changed but the dairy product didn’t.
Citation needed.
>If yoğurt was oxygala Greeks would continue to call it oxygala.
That's not how language always works. Greeks have adopted foreign works for things they themselves had in the past (with their own words), and also use ancient Greek words for things that were imported much more recently. Even something as basic as water has changed name from ancient Greek times.
Words have complex itineraries and adoption cases. Sometimes a newer foreign word is adopted because that's when a product gained more popularity (and the older one has been forgotten), or merely because it's thought as "more modern" to refer to it by that name.
There are is even "reborrowing", when an ancient words is re-imported, centuries or millennia later, after it has been adopted and slightly altered abroad.