For learned Muslims, this phrase is very connected with Islam. They are taught the proper etiquette regarding the salām, and among them are rules regarding non-muslims:
1) One should not initiate the salām towards non-muslims, and
2) If greeted with salām from non-muslims, the answer should be "wa alaykum" (and upon you too).
Among the reasons I learned is: because as-Salām is one of the names of God Almighty, you shall not invoke his holy name upon a non-believer.

show me where in the quran mentions anything about "as-salamu alaikum". in the quran it is either "salamun alaikum" or "salam" only. Allah forbids addition and invention in religion. It is called Bid'ah.

Dan mereka membuatkan sebahagian daripada apa yang Kami merezekikan mereka kepada apa yang mereka tidak tahu. Demi Allah, kamu akan ditanya mengenai apa yang kamu mengada-adakan Q16:56

And they assign to what they do not know a portion of that which We have provided them. By Allah , you will surely be questioned about what you used to invent. Q16:56