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.

That is because you follow a religion that has only the al-Qur'an as the guidance, whereas majority of Muslims follow the Sunnah on top of that.

(This is going offtopic.)

And what makes you think that particular word in that verse refers to the Hadith that's collected by Muslim scholars?