Well of course he is. And he should see it as an honor. How many lowborn omegas can say that they were chosen by THE House Targaryen as their family omega? Who gets to take care of their pups and sate the needs of their alphas. It is a privilege and a fitting way to correct his earlier behavior. So unomegalike, wanting to be a knight. Where do omegas get such silly ideas? His place is in the Keep, docile and bred. He's lured in with a pup, Egg, who was unwitting in making Dunk so attached to him but he also doesn't want anything bad to happen to Dunk so he urges him to just accept the offer so that they can be together in the Keep. Egg has always wanted to be in an omega's nest where he's coddled and he will soon present, he needs the education that only demonstrations will provide, as will his sisters.