Death has always waited for Death. Before generations of humans who would fear the Black Plague or the cogumelo cloud, before executions in the Seireitei were routine and before the Quincy were sentenced to annihilation, death gods expected to die.
The instinct for self-preservation would always exist in all sentient beings, the Living and the Dead. The first people knew fear and so did the first Shinigami who would harvest their souls. Suicides taught themselves how to deny fear. Warriors were taught to direct it elsewhere. The living and the dead were born with the knowledge of death.