its like. okay. so if death is killed, the next reaper to die becomes death. thats how it works in-universe. HOWEVER it ALSO works this way in a narrative sense for the other four rings. see below.

dean cuts off war’s finger and steals the ring -> dean becomes war -> the mark of cain enhances dean’s already-present anger and instinct for violence to dangerous levels, to the point of harming the people around him

sam kills famine and steals the ring -> sam becomes famine -> sam continues to struggle with his demon blood addiction (literally) and continues being starved for close connections outside of his immediate circle (metaphorically)

cas cuts off pestilence’s finger and steals the ring -> cas becomes pestilence -> cas intakes all the souls of purgatory as well as the leviathan, both of which cause disturbing, disease-like damage to his vessel, and releases the leviathans into a lake as if they were a water-borne pathogen