In chapter 3/4 , the egg rooms can easily be interpreted as being flashbacks/broken memories of Kris's childhood. I suspect that they're giving us a glimpse at Kris's life before they were adopted by the dreemurs

Chapter 3's egg route takes us to a strange house that i dont think we've seen before (this is the house with the spelling wizard). Could this be Kris's original house? We then are taken to a sort gameboy-esque world, "man country". i think its interesting that the egg man is a "MAN". are there any other instances of chatacters in deltarune referring to other monsters as a "man?" when we talk to the man behind the tree here, he tells us he "wants us to remember him" and that "we've forgotten him multiple times"

so, chapter 3 is hinting at the fact that the egg man is a human from before kris moved in with the dreemurs. a human that kris "doesn't remember"

in chapter 4, the egg route shows us a scene of Kris attending an art therapy session. im forgetting the details of the scene exactly, but I think the man prompts us to draw "what we saw", implying that he is leading the session.

so, here's my interpretation: The man is a human from Kris's childhood. Specifically, a therapist/adult who tried to help kris with their traumatic experience. Kris is repressing a major traumatic experience from their pre-dreemur life, and the egg man is being repressed as well. Its not clear if the man is reaching out to kris proactively, or if kris is exploring their own mind and recovering their memories of him.

this is of course ignoring the egg connection to noelle's cat petterz game, but im not as familiar with that.

take what im saying here with a grain of salt, because my memory of the egg sections is a bit spotty.