I initiated a power challenge against Player A with Robert Baratheon (from the Grand Melee), and Player B (Martell) added his character as a defender. Player B then played Burning on the Sand to kill the challenge.

Burning on the Sand:

If an opponent would win a challenge against you, cancel the determination of the challenge winner. No one wins the challenge.

Is this legal? Is the challenge "against" Player B if it is initiated against Player A? Or, because they must satisfy claim, is the challenge against all opponents?

I am pretty certain that only the player that the challenge is initiated against, redirected to, or chooses to Support a player whose title they support is considered the "defending player". While winning the challenge will hurt Player B as much as it will hurt player A, that does not mean Player B is the defending player. Otherwise, wouldn't The Red Vengeance make the player that is affected by claim the new defender?

The answer is actually right there in Robert's text:

"While Robert Baratheon is attacking alone in a POW challenge, each opponent you are not attacking may declare 1 eligible character as a defender. If you win the challenge, instead of the normal claim effects, each opponent must satisfy the claim effect of the challenge."

So you are not considered to be attacking the other players, and you are not considered to have won a challenge against them. The effect requiring all opponents to satisfy claim effects is a replacement of the original claim - still won against the original defender.

So if A attacks B, C cannot play Burning in the Sand, even if the attacker is Robert because A did not win a challenge against C (making BitS ineligible).