There are a few different customs. I asked 2 different Rabbanim about this after noticing that my Siddur said to say "Shalom Aleichem" to 1 person 3 times and not finding a satisfactory explanation for the difference in the Tur, Shulchan Aruch or Mishna Berura. One of the Rabbanim came back a few weeks later and told me the following: (He said he looked in a Sefer that explained different Minhagim but I don't recall what the name was.)
There are 2 different Minhagim in regards to "Shalom Aleichem"in Kiddush Levana, both of which are valid. The most common practice is to say it 1 time to 3 different people, which makes sense as this is the normal way to greet people. The other Minhag is to say "Shalom Aleichem" three times to one person. Incidentally I recall listening to a pre-recorded Shiur on YUTorah.org in which Rav Schachter said the 2nd Minhag was Rav Soloveitchik ZT"L's custom.
As to what you saw, that wasn't mentioned. If it is a Minhag, it must be rooted in a desire to say every word, but a concern for Shome'a K'Oneh, meaning that if someone simply answers "Aleichem Shalom", they've essentially said "Shalom Aleichem" to however many people prior to saying it themselves. I don't know if that applies to Kiddush Levana or not though.
As a total side point, if you look in the Artscroll Siddur with the English, it says to say "Shalom Aleichem" once to 3 people. The Hebrew Siddurim says the inverse - 3x to 1 person. This is the true for both Nusach Ashkenaz and Sefard - I checked.