The creature is possibly named Trico because it's in the third game of the ICO series, completing the trio of ICO games.
The ending reveals that the two times Trico got controlled into eating the boy, he didn't do so out of malice or hunger, but rather the mind control device telling him to collect the boy.
It seems odd that every time Trico is controlled into eating the boy, he faints and throws him back up afterwards. But then the ending shows that once the Master of the Valley is destroyed, all the Tricos wake up from their control and either fall from the sky or the tower as they apparently faint from the shock, likely the same thing that happened to the main Trico those two times in the game.
The Stinger reveals that Trico had a baby. His mate was probably the darker Trico he and the boy occasionally encountered throughout the game, considering the loss of the helmet reverted her back to normal and she was shown to still be alive and in the valley at the end. (Alternatively reverse the genders.)
Word of God confirms the dark Trico is the opposite gender of the main Trico, possibly in support of this interpretation.
Trico's gender is left ambiguous, but at one point the boy uses a masculine pronoun to refer to it. He's probably right; Trico's feathers are light grey, and his ears are bright blue, whereas the other main trico encountered is dark. In most birds, the males are the more colorful members of the species, while females are colored so as to blend in with their nests. If trico nest in dark caves as the ending shows, it would make sense for the females to be nearly black, while the males are lighter and brighter.
At the top of the White Tower, it's shown that the Master of the Valley rewards the tricos with barrels for successful deliveries. However, it only ejects one barrel for a group of three tricos, causing them to fight over it. This is probably to foster more aggressive behavior and competition so that the tricos will bring more and more people in hopes of being able to get a barrel. This also is a possible indicator of why our Trico latched onto the boy so quickly; he's been spoiling the critter with treats the whole game, on top of being kind.
Why is the Nest the way it is? It's certainly not any sort of practical structure for human beings to inhabit. Then you realize - the ruin isn't designed for human beings at all. It may have been built by humans long ago, but it was probably designed by an Eldritch Abomination explicitly for the use of it, its Hollow Men and the tricos - why your very large critter can fit through most passages and why there's absolutely No OSHA Compliance against death by falling anywhere in the ruin.
You often have to find food for Trico, but the boy himself never eats.
This makes at least some sense. The game takes place over a short time frame, and human beings can go quite a while without any food if they have to. Bigger animals like Trico, unless they've got a lot of fat reserves, can't, and the barrels the Boy feeds him aren't very big, so he needs a LOT of food to perform the feats he has to.
Pay attention to Trico's mouth and it's apparent that the creatures have no teeth, which begs the question how Trico gets so badly hurt every time another Trico (or several Tricos by the end) bites him. Granted the ending Tricos may also have clawed him, though they still appear to be primarily biting him.
Most toothless animals have rather hard mouths, either a full-on beak as in the case of birds or a sharp ridge in the gums. Trico's lips are mobile, so it's apparently an internal structure. It would be difficult to eat if you couldn't tear your food apart. And as anyone who has ever been goosed by an angry goose can tell you, you don't need teeth to do serious damage.
A cutscene reveals that Trico lost its wings in an accident, but for some reason the dark Trico that sometimes appears throughout the game also has stunted wings. It might have been to not give away that Trico used to have, or is going to have, full wings at some point. Another guess is that the dark Trico got its wings clipped by the guards to act as a sort of guard dog in the valley.
Had it not been for single, lucky stroke of lightning, the Boy would have been fed to the Master of the Valley just like all the others before him.
When Trico and the second beast are freed from the master's control we see that their kind are friendly, intelligent creatures when not threatened or protecting those they care about, and after the master is destroyed we see all of those other griffins possibly fall to their doom, however Trico survived fall from similar heights so let's hope at least some of them lived to enjoy their new freedom.
Trico is terrified of those glass eye symbols. Since this is true even when he's not under any mind control, it's highly likely he was conditioned to fear the sight of them. This might have been accomplished a few different ways, but the easiest way to do this would be to teach him to associate the symbols with pain - possibly from a young age.