Taking these to be the symbols of Islam is a wrong assumption. This wrong notion is normally based on what is known of Christianity i.e. no cross no Christianity. This does not apply to Islam.
The crescent-moon and star have no religious implications nor are they part of the fundamentals or basic teachings of Islam. Their origin can be traced back to the Ottoman Empire (Turkish Caliphate) which adopted them for some historical reasons.
But there is no basis for it in the Qur'an and Sunnah. However, it is considered as Mubah i.e. tolerated most especially as it can be useful in identifying mosques in some areas where the architectural structures of a mosque and a church appear the same.
The symbols sometimes arouse confusion among non believers as they may mistakenly associate Muslim with the practice of worshipping the moon God. Allah almighty has denied this 1400 years ago through the revelation in Quran which says;
"And from among His Signs are the night and the day, and the sun and the moon. Do not bow down (prostrate) to the sun nor to the moon, but only bow down (prostrate) to "Allah" Who created them, if you (really) worship Him.”[Holy Quran 41:37]