This symbol became affiliated with the Muslim world during the time of the Ottoman Empire. This had been a symbol of the city of Constantinople, and when the Turks conquered it (in 1453), the adopted the symbol of the city and renamed the city Istanbul. Legend holds that Osman, the founder of the Ottoman Empire, had a dream in which the crescent moon stretched from one end of the earth to the other. Taking this as a good omen, he chose to keep the crescent and make it the symbol of his dynasty. For hundreds of years, the Ottoman Empire ruled over the Muslim world. After centuries of battle with Christian Europe, it is understandable how the symbols of this empire became linked in people's minds with the faith of Islam as a whole.
There are several Muslim countries that currently feature the crescent moon and star symbol on their national flag. But many Muslims reject this as a symbol of Islam. The faith of Islam has historically had no symbol, and many refuse to accept what is essentially an ancient pagan icon. It is certainly not in uniform use among Muslims.