I think as Nudnikit said, you are probably part of a stereotype. Many times people react to situations in interesting ways because of connections in their mind to many other things. In their view, by giving you a $10 present they are giving in, they are surrendering. Here they believe you are supposed to be able to support yourself and now they should, with their own hands, contradict themselves? You are right that it is not a real argument, you actually agree to them and it is a new circumstance that came up. However, in their mind, it would mean bending, undermining their values, being taken advantage of, and who knows what else.
I once heard that Rav Pam z”l said a Mashal. If you are in bed and hear the sink drip you won’t be able to fall asleep, and will have to get up to close it. But, when you hear the rain drip, perhaps even louder than the sink, you have no problem falling asleep (it might even be considered a pleasant sound). The difference, he said, is that you can do something about the sink but not about the rain.
IT makes you upset to hear them talking about how they have money, and when you see them giving other siblings items of less importance. When you put it out of your mind, that it is not something that will be coming your way and that money has nothing to do with you, that should ease the feelings.