I know they save a character from being killed or discarded, but in the manual, it seems to me that they only can do that when it's by an EFFECT of a card.

Can they save a character from being killed, let's say, if the opponent wins a military challenge, and then instead of killing them, the players can just discard a duplicate?

Rulebook, Page 12:

"A player can discard a duplicate attached to a unique character in his play area to save it from
being killed (see “Save Responses” on page 13). "

Says nothing about "only when killed by a card effect".

So, yes, you may discard a duplicate to save a character from being killed for claim.

Hello! Would you say this also works when Valar Morghulis is revealed? If one character has a duplicate at that time, does it mean that the duplicate is discarded but the characters survives the plot?

Yes.