Oh, I know what they say, and it was by far the "greatest" hotel of its time and many people share the same nostalgia as your family, however:
1) In Turkey the "Star" system is done by the Ministry of Tourism. They had no "Star" system until the 1970's.
2) Until the mid 80's internationally "4 Star" was the top rating in pretty much every country which had a "Star" system.
So while the Istanbul Hilton was way ahead of its time, more than doubled the "top" rooms in the city, etc etc. Any talk of it being the first "5 star" hotel is pure marketing speak (probably written by someone in their 20's), as no such designation actually existed either internationally or in Turkey for another 30+ years.