I am not sure this is so much an issue of adolescence and Ellie feeling an awkwardness in his relationship with his father; I think it is more an issue of tolerance of the impositions of the government and the shame of having to mark oneself as somehow different outwardly; it is degrading to have to point oneself out in the crowd, and the results of wearing this star was that you would be treated differently. This is a fate that Ellie felt was already bringing a sort of lifelessness to their existence. By saying "Of what then do you die?" Ellie means, perhaps naively, "how could it be worse than identifying me in this way, humiliating me, and limiting my rights? I am already feeling dead from this", whereas Ellie's father merely accepted the reality of the new law, because there seemed to be no use getting upset over something they could not change. After all Ellie's father was right: you don't die of wearing the yellow star, literally, you are simply marked for death. It's a figurative precursor to death, the beginning of a long road towards eventual death, most likely physical, and definitely, inescapably emotional and mental.
Eliezer possess a complex relationship with his father. On one hand, he experiences a level of emotional distance from his father, as akin to many adolescents do in regards to their parents. In the rise of Nazi Germany, it can be presumed that some of this view of detachment from parents caused by embarrassment of a lack of respect can be enhanced because of the degrading actions of the Nazis exhibited towards their victims. When the young Eliezer says that above quote and remarks of what his "father died of," it is an indication of the distance felt between an adolescent and his father, enhanced by the Nazi treatment of Jewish individuals. We see this often in the work. In the various camps with his father, there are moments when he is ashamed of his father and seeks nothing more than to be spared his embarrassment. Having said this, the complexity in the relationship is that Eliezer exhibits a great deal of loyalty towards his father. At different points in the work, Eliezer takes care of his father, risks his own well being in staying with him, and runs along side him during the evacuation to Buna. The awkwardness of adolescence that causes moments reflecting a lack of respect- such as in the statement above- as well as moments that show loyalty was only enhanced through the challenges and unimaginable difficulties in the period of the Holocaust.