Why do we say Shalom Alaichem to one person if Alaichem is plural?
Is it saying you and all of the Jews? Is that why it's plural?
I have a good friend who says "Shalom Alecha" to me during kiddush levana. In fact this is the nusach brought in the by the Rama in the Shulchan Aruch (OC 426:2) and it's also in the Tur (OC 426). (Note that the Bach there comments that this is the proper version of the text.) I always smile and respond "Aleichem Shalom."
The greeting Shalom Alecha, in the singular form, appears in dozens of places in the Talmud.
I suspect that the usage of Sholom Aleichem stems from the famous song Friday night, where we are addressing multiple angels.
In addition, it is considered respectful to address elders using the plural form, and this usage may have become more commonplace since we usually use the words Shalom Aleichem when we are being a bit more formal.
I heard in the name of the Hafes Haim (and it is brought down in Ben Yehoyada Berachot 3) that one has two angels that are with someone during the week, and two different ones that come on Shabbat (that's why we say Setechem LeShalom).
It could be, that the person who is greeting you is also greeting the angels.
Like the "royal we" except giving respect to others instead of ourselves
This is clearly as a show of respect. The second person plural can be used to show respect in Hebrew. Also in Arabic an almost identical greeting is universally used, "As-Salamu Alaykum". Seems pretty clear that this is something that goes back quite a long way in Middle Eastern culture including Judaism.
The Wikipedia entry for Shalom aleichem says that the reason for the plural form is that one greets both the body and the soul.... (no source is given for this however)
Avi Ron wrote an article in the Hakira journal on the topic of greeting 3 people during kiddush levana, and he says that when Masechet Soferim 10:2 discusses the specifics of kiddush levana it doesn't use alecha or aleichem, it just says to say "Shalom" to your fellow 3 times. He then says that Machzor Vitry,4 Rokeach,5 Or Zarua,6 Shibolei HaLeket 7 and Tur8 all write Shalom Alecha, while Shalom aleichem is written by the relatively recent halachic works: Eliya Rabba,12 Shulchan Aruch HaRav13 and Aruch HaShulchan
why did the form of the greeting itself change from the singular to the plural? This has to do with the development of the use of the plural form in Hebrew to denote respect. As noted above, the Talmud does not record the use of the plural form to greet an individual, even an individual who is due great respect, like a teacher.31 Use of the plural form as a sign of respect, called vouvoiement in French, is not originally found in Hebrew or Aramaic, but rather was borrowed from French and German.32 R. Yair Chaim Bachrach (d. 1702) in his Mekor Chaim writes, “ ‘Shalom Aleichem’ that is commonly said is not correct for an individual in Hebrew, and the error comes from foreign languages.”33 He notes that the Rema preserved the original Hebrew expression ‘Shalom Alecha’ when writing about Kiddush Levana, and that R. Yosef Caro used the singular Hebrew form when writing about greeting a rabbi.34 Despite this, over time the plural form of greeting became accepted as appropriate. Various reasons were given to explain the special spiritual significance of greeting in the plural form, for example, as a greeting to both body and soul.35 Eventually the foreign influence was forgotten and the plural form became normative. By the 19th century, R. Yechiel Epstein simply states that “in ancient times people would always speak to each other in the second person, saying ‘Shalom Alecha Rebbi,’ but today we address an honored individual in the plural form and say ‘Shalom Aleichem.’”36
Refer to the link for footnotes: http://www.hakirah.org/vol%207%20ron.pdf
I once heard that at Kiddush Levana we say Shalom Aleichem - in plural - since we do not want to make the same error the moon made when it complained that 2 kings can not serve at the same time, therefore we say plural to include the moon and the sun. However this would not answer why when you meet a fellow on the street do we say it in plural.
Way back when, a few decades ago, in Yeshiva Ketanah, I was told that we are accompanied by two angels, which is the reason for the pluralization of Alecha.