We have a military challenge and the claim is 2. The defender has 2 characters. One is with a duplicate.

Can the defender discard the duplicate and kill that character with duplicate, or must he discard the duplicate and kill the other character?

You must choose your claim targets at the same time. So you cannot choose the same character twice.

Core Set rulebook, pg. 22:

"High-claim Military Challenges
If, during a military challenge, the defending
player loses a challenge, and the attacking player
has a claim of 2 or higher, the defending player
must kill the required number of different characters,
all at once. Thus, the defender may not
choose to kill the same character multiple times,
even if that character could be saved."