If I were to win a challenge with both Ser Gregor Clegane and Euron Crow's Eye and my opponent discarded two characters during the pillage resolution, would I be able to trigger Ser Gregor Clegane's reaction multiple times?

No.

Ser Gregor specifically says, " Reaction : After Ser Gregor Clegane discards a character using pillage...". So only his own Pillage effect meets the triggering condition requirements for his reaction.

If you have multiple characters with Pillage, you have to indicate, "Discard the first for X, and the second for Gregor." so you know which card applies to which character. This is no different than having to say, "Will uses stealth on X and Ghost uses stealth on Y" when you attack with multiple stealth characters, and one or more of them have reactions to assigning stealth (like Ghost).

That's what I assumed but it didn't say anywhere in the RRG that each pillage effect is assigned to a specific character, just that you discard one card for each character. Thanks again for the quick response!

Card text takes precedence over the basic rules framework. If you think about it, that's the entire point of having text on cards at all.

If you have multiple characters with Pillage, you have to indicate, "Discard the first for X, and the second for Gregor." so you know which card applies to which character. This is no different than having to say, "Will uses stealth on X and Ghost uses stealth on Y" when you attack with multiple stealth characters, and one or more of them have reactions to assigning stealth (like Ghost).

So after replying to you yesterday I went back a re-read pillage and stealth yet again and, while I trust your authority on these things, I'm still not convinced this is the way it works.

In the RRG the rule for pillage reads:

Pillage is a challenge resolution keyword. After a player wins a challenge, for each participating character with the pillage keyword he or she controls, that player may discard 1 card from the top of the losing opponent’s deck.

To me this says that the player has a pillage count at the end of every challenge, and for each participating character the player gets to discard one card, rather than assigning each pillage discard to each participating character.

Pillage says specifically that the player is the one discarding a card from the opponent's deck, which is important for my next point.

Ser Gregor Clegane's text reads:

Reaction: After Ser Gregor Clegane discards a character using pillage, place that card in its owner's dead pile. Then, you may choose and kill a character with printed cost equal to that card's printed cost.

Based on that I'm starting to believe that Ser Gregor can never actually trigger his own ability since the card/character 'Ser Gregor Clegane' is never discarding a card using pillage, but rather the player is.

All that being said here are my questions:

1. ​Can Ser Gregor Clegane trigger multiple times in a pillage resolution involving multiple characters with the pillage keyword?
2. If he can only trigger once, and the reason is because pillage has to be assigned to a character rather than come from the player, where is that stated in the rulebook?
3. If pillage is coming from the player and not the card/character, how does Ser Gregor Clegane's reaction even trigger in the first place?

" Pillage is a challenge resolution keyword. After a player wins a challenge, for each participating character with the pillage keyword he or she controls, that player may discard 1 card from the top of the losing opponent’s deck."

You are emphasizing the wrong part of that. Try:

"Pillage is a challenge resolution keyword. After a player wins a challenge, for each participating character with the pillage keyword he or she controls, that player may discard 1 card from the top of the losing opponent's deck."

When you emphasize that, you get that the character with the keyword has a direct individual action. "For each" indicates a one-to-one correlation between the character with the keyword and the card that leaves the opponent's deck, so when you have multiple characters with the keyword, it's not "Pillage equals 2, so discard 2 cards." It's "Pillage for Gregor, then Pillage for Euron." The same goes for stealth. It's not a question of "Attacking stealth equals 2, so I'll target 2 characters." It's "Stealth for Ghost, then Stealth for Will."

Further, we know this is the more accurate emphasis and procedure for these sorts of keywords because, as you say, abilities like Gregor and Ghost would not work if Pillage and Stealth were always treated as a single, general, "collective" resolution.

With that in mind, the answers to your questions are:

1. No. Gregor can only ever react to the discarded card that correlates to the use of his own Pillage keyword, no matter how many other Pillage characters are in the challenge.

2. The definition of Pillage - pay attention to the "...for EACH participating character..." wording.

3. Pillage is coming from the game rules, not the player or the card, but the game rules create a one-to-one correlation ("for each") between the character with the keyword and the card that is discarded. That's why Gregor works in the first place. Otherwise, FFG printed a card (several of them, in fact) which do nothing -- and we have to assume that an interpretation of a card or the rules that results in "nothing can ever happen" is incorrect.