Is 'each other' used correctly when talking about sets of more than two people?

It's correct in a conversation between two people. When addressing two or more people, use the phrase 'one another'.

"We have a christian duty to serve each other. Is this statement correct**?** Because each other is used for two persons and one another for more than two persons.

This seems to be an invented "rule" that has no basis in fact.

The OED gives "each other" as synonymous with "one another" and remarks:

each other pron. used as a reciprocal pronoun (as object and in the genitive) = one another. … Some commentators on usage restrict each other to two parties and one another to more than two, but such a distinction is seldom found in actual use.

My emphasis

This seems linked to the concept of "clusivity", which is missing from English.

Broadly speaking it is the distinction of whether the speaker and/or the addressee are included in the group referred to by "we".

So, if "we" is meant to include the speaker and adressee, "each other" seems appropriate. If "we" is meant to exclude the speaker, the adressee or both, "each other" would be lacking the "other".