If I win (or my opponent wins, it's the same) a claim 2 intrigue challenge can I activate twice the response of the Guardian of the Way, or it activates ony ones?

I think I can activate twice since the response can be activate each time a card is discarded from a player's hand.

Yes, the response can be triggered for every singular card which is discarded from hand. Also keep in mind that it read´s "from a player´s hand", so cards that are dicarded from your hand can also trigger the repsonse.