I've read this story several more times, and I may well be wrong in thinking that the narrator is an over-the-hill robot boy. There are a few details that make more sense if he is a mature human male. However, I can find nothing to suggest that Julianne's parents are robots. In fact, in a human-engineered society where human children seem to be at a premium, I very much doubt that robot couples would be allowed to foster human children, even if robot couples even existed, a proposition for which I can see no justification in the story.
But for use of the plural-possessive pronoun "our", referring to the narrator's back yard, we might suppose that he lived alone. He makes no specific mention of another person or persons in his house, other than to state that there are no children in it. The most logical assumption is that he has a wife and that, like the Robinsons, they are either unable or unwilling to have children, which is why he could not blame them for being in the same situation he was in. But he could -- and probably did -- blame them for acquiring a robot child, just as he later became incensed when he learned that Julianne's parents were getting her a robot dog in reaction to the robot girl getting a real dog.
Despite the fact that most houses in the neighborhood housed no real children, robot children did not dominate. There were three robot children in Roberta's first-grade class and seven human children, of which five of the latter group were boys. If Julianne's best friend was a new(*) robot, that says more about her than it does Roberta or the society she lives in. Sadness is a part of the human condition, whatever its cause(s). Julianne had the opportunity to make friends with the other six human classmates before Roberta ever came on the scene, but she didn't, for whatever reason. (Part of her sadness may stem from being made to feel stupid relative to the obviously much more knowledgeable robot children, who are in school only to conform with their owner's fantasies.)
Rover instinctively understood that Robber was a thing, not a dog. It is unclear, however, that six-year-old Julianne realized that Roberta was a mere robot.
Is there any significance to Wolfe's use of the British spelling of "grey"?
[(*) It took me several readings before the point of Roberta's "little laptop in a shiny new bag" struck me. The first time the narrator saw Roberta was on her way to school in **May**, near the end of the school year, not the beginning. Apparently, Roberta hadn't been around in April.]
-Roy