This is going to be a weird comparison but I remember a lot of RPGs (both "character" ones and Monster ones) where you could link-interact with friends and then might get the chance to recruit their characters or monsters to your roster, straight out of their party. This was to the point that it was the only way to complete a Bestiary through either Version-exclusives or Duplicating rare monsters for Fusion chains. Since Pokemon, for multiple reasons, probably wants to discourage that as pseudo-cloning, maybe the Eggs are a compromise.
So if I really like Mark's Perfect super-strong Charizard Nidoking, but he doesn't want to trade it to me, then I can get an egg while picnicking in his game that will hatch into a Nidoran that is essentially a baby version of his (IV inheritance and Egg moves mainly) to raise up myself. Doesn't have an overt "given by X" world-role, benefits the Player Interaction aspect, and organically keeps the system to non-Breedable Pokemon (so no worries about stacking Legendaries or Mythicals in the Pokemon-version of that system)
While true, specifying Eggs in the post about group interaction for Picnics in what seems to be the latest "Multiplayer Hub" (Union Room, Join Avenue, PSS, Festival Plaza) does make me wonder if there's a specific interaction or behavior here compared to Solo, as a way to further encourage socialization.