Fun fact: the moon and star symbol was actually applied to Constantinople as far back as antiquity when it was named Byzantium. My understanding (I could be wrong) was that it was associated with the popular cult of Diana.
It stayed with the city even after it became Constantinople, center of the Eastern Orthodox Christian faith, and later was applied to the Ottoman empire and Islamic world, hence the presence of the Moon and Star in so much Islamic iconography and flags today.
Again, that's the history as I learned it, and it was a brief one from the otherwise in-depth Constantinople episode of "Fall of Civilizations" (good History podcast on YT, btw). If anyone a dispute regarding this feel free to mention it.