We have a situation where a player has the Fury of the kraken plot revealed. after winning a military challenge and the target from Fury is chosen, can the opposing player choose the same character as the target of the challenge loss?

Skatell said:

Actually, you have the order backwards there.

When a challenge is resolved, 4 things happen before anything else can take place: the challenge winner is determined, claim is settled, power for an unopposed challenge is awarded (if applicable) and power for Renown is awarded (if applicable). These are the basic steps in challenge resolution.

Passive effects to any of those 4 things must wait until all 4 of them are complete. So even though "winning the challenge" happens before "settling claim" in the order of challenge resolution, you still have to go through the entire challenge resolution before any passive effect activated by winning (or losing) the challenge is activated. In short, you do not choose the target for Fury until after the character is killed for the lost military challenge claim.

All that aside, no - if a character is killed or discarded by one effect, it cannot be killed or discarded a second time.