I think Doom Slayer must have taken that rabbit's tail from Doomguy's dead corpse, he found Doomguy's dead body, gave his silent respects to the man, and took Daisy's tail from Doomguy. Besides, Quake Champions and Doom 2016 are in different canons, Doomguy appears as a skin for Doom Slayer, along with the Doom Marine from Doom 3. Besides, We have not seen Doom Slayer's face, but we have seen Doom Guy's face, as well as the unnamed Doom Marine's face.