I think horses get more "confident" with any kind of training because they learn that they can be in control of some aspects of their environment. When I put my left leg on and my horse shifts her weight to move to the right, I release the pressure (and sometimes reinforce with a "good girl" or wither pat)...she learns that she can "control" the stimulus - it will go away with certain behavior. That gives her confidence, of a sort. Is it confidence in her abilities as a dressage horse? Probably not. Is it confidence in her newfound athletic ability? Probably not. Is it confidence that I'm a reliable partner and not going to kill her as her instinct says I will? Maybe.
Now, I'm not sure my mare goes out in ther pasture and says to her pasturemate, in effect, "Hey, big boy, notice how my butt is getting rounder and more muscular? Don't you love it?" I'm not going to go THAT far with the word "confidence."
I absolutely think that the more you train a horse, the smarter they get, the more they know, the more in tune and the more they respond to things
My 5 yr old gelding is more confident now than he has ever been. He was a stallion until 3 1/2 and I thought he was confident as a stallion, but that was nothing compared to the managed confidence that maturity and training has brought him.
I see it especially as our younger horses develop in their training.
I agree they gain confidence. As they advance they are not guessing at the answers to the questions we are asking. A certain aid means they respond in a certain way. With this comes calmness and inner confidence.