2016 and Eternal are still in my backlog, but I remember discussions about Eternal showing the hallucination of a bunny, and it was a hint that it was in fact the same doomguy: OG ends with the reveal that doomguy’s pet bunny on Earth was murdered and beheaded during the invasion by the demons (that ending screen with a bunny’s head on a spike). It’s the whole reason doomguy is mad at the demons, and he’s still pissed in 2016 and Eternal.

It might be mostly player deduction and there’s no official lore about it, but the bunny is definitely a hint or at least an easter egg. The wiki says the pet bunny was called Daisy.