"Mon peuple" refers to only one (group of) people....my people. "People" is a word which refers to many individuals but it is spoken of as a single thing. I would say it behaves the same way as "le raisin" refers to "grapeS" and not "a grape". One could say "the peoples of the world" if referring to the many different "people" groups which comprise the entire population. But in this sentence, that is not the case. So, "You are my people" becomes "Vous êtes mon peuple", not "...mes peuples."
I think what's significant here is that the use of "un peuple" as a collective noun is more common than the parallel use of "a people" is in English.
Probably this is because of the fact that "people" is also the usual plural form of "person". You can say, for example, "The French are a proud people", or "The French are proud people"; there is a slight difference in meaning, but not much.
In a sentence like the one we have here, "You are my people", the word "people" could technically be either the collective or the plural, but I think you'd find that people generally hear it as the plural.
I base that assertion on the observable fact that in general, Anglophones want to translate it as "mes peuples", or "mes peuple", not "mon peuple".