I rescued Billy, my Aussie Terrier, when he was abandoned at a dog park at about 2-3 years of age, 6 years ago. He's the only rescue I've ever had, and he's the most loyal, devoted dog I've ever had. Do adult dogs know they've been rescued, and show it?
Dogs, in varying degrees, are driven for social interaction, a comfy and safe shelter, food, play, and other things that a home provides them. They don't understand the concept of being adopted or rescued, or feel grateful, in the human senses of those words.