The answer is that Dusty is not identical to his physical parts. Certain things are true of Dusty Crophopper that are not true of any of his constituent parts. Wings aren't brave, but Dusty is. An engine isn't determined to never give up, but Dusty is. There are also things that are true of his physical parts that are not true of his identity. Dusty does not rust, but his parts do. Dusty does not need oil, but his engine does.
Ironically, the intuition of the antagonist plane is not based on something we know to be true of airplanes in real life. In reality, if you replaced all of the physical parts on an airplane--even if you kept the same engine--no one would say that it is still the same plane. So, what is the basis in reality for the antagonist's claim that identity is not merely a physical property? Just like other Disney movies, which feature talking toys, cars, monsters, and animals, the characters are based on what we know to be true of human beings. We know intuitively that human beings are not identical to their physical parts. When a boy breaks his arm, he doesn't say, "I am broken." No. He says, "I broke my arm." There are things that are true of him that are not true of his arm and vice versa. That is why it is his arm.
If Dusty Crophopper were identical to his physical parts, then bolting on new parts would change who he is. That is true even if he has the same engine, since there are things that are true of his engine that are not true of Dusty and vice versa. If human beings were identical to their physical parts, then I would not he same person as the 4 year old child in my family photo albums. I would not really have an identity that persists from one moment to another at all. Just because I have the same brain doesn't change that. There are certain things that are true of me that are not true of my brain and vice versa. My brain does not love my wife and kids, but I do. My brain does not intended to go to the store this evening, but I do. I am not a moist lump of grey matter, conducting electrical impulses, but my brain is. I do not weigh approximately 1400 grams, but my brain does. Dusty is not his engine, and I am not my brain.
Just because they are not identical does not do anything to undermine the crucial correlation between Dusty and his engine, or between me and my brain. Dusty's engine didn't purpose to join the Wings Around the Globe air-race, but he couldn't have made that decision without his engine. If it had been totally up to his physical parts, he would have kept crop-dusting since that is what he was built for. In fact, that is the underlying point of the whole movie--you are more than just a physical robot programmed to do follow a particular function. You can make your own decisions and accomplish great things. But that would not be true if Dusty was identical to his physical parts. In the same way, my brain didn't purpose to seriously overanalyze a children's movie, but I could not have done it without my brain. However, I were identical to my physical parts, it would not matter that I did, since I would ultimately be determined to follow my biological programming. My thoughts would be no more meaningful than the explosions that take place in a real-life airplane engine. Disney's "Planes" highlights our shared intuition that this is not the case. We are not identical to our parts.