His disdain for the reaper's curse notwithstanding, Eizen doesn't shy away from referring to himself by the moniker. If anything, he seems to relish any chance to use his self-proclaimed title and feels slighted when underestimated. His obsession with this sobriquet likely stems from his origin story as a pirate and the influence of his captain, Van Aifread.
When he first set out to lift the curse endangering his sister, Eizen felt that his life was bound to a fruitless quest as the days passed with no progress. Aifread, who later boarded the ship that Eizen was on, encouraged him to embrace his curse and not be enslaved by it. Inspired by the gusto with which humans carry out their relatively short lives, Eizen has since devoted himself to living each day to the fullest.
Eizen's wanderlust and subsequent piracy are the height of his emotion. By choosing to live, he has given up on looking for a cure, damning his sister to familial solitude by extension. He states that he loves to explore, to dance with death in a thrilling rush of adrenaline as he plunges into the unknown. When pressed to tell Edna the truth about his being a pirate, he cowers at the suggestion, fearful that she would disapprove of the only way he knows how to "feel alive."
Hilariously, his love of antiques is so great that he would even steal from Edna's secret gald stash to buy a painting that has moved his heart beyond words. One might say that they are his greatest vice.
Who is the Reaper? A pirate who doesn't fear death.Eizen is less concerned with surviving and more preoccupied with making every day count. Observe his second mystic arte's dialogue: "I don't need a tomorrow! In order to kill you, this is checkmate!" A life that's not of his choosing is no life at all, insofar that he actively endangers himself: Whereas most malakhim steer clear of humans, he expresses his fondness of mankind and mingles with his fellow pirates at the risk of taking on a detrimental volume of malevolence. As far as Eizen is concerned, becoming a dragon is no chip off his shoulder, so long as he's led a meaningful existence before the inevitable.
Thus, while Eizen's image is associated with death, his character celebrates the gift of living to counter the connotations of dying. He owns his title, because it is symbolic of the moment that he wrestled control of his path from his curse.