The first two robot stories (again in the fictional chronology - in actual terms, A Boy's Best Friend appears much later - Robbie is, I believe, the first Robot story written by Asimov) are incredibly similar involving a young child becoming attached to a robot (robot dog or nursemaid) in preference to the "real thing", much to the parent's chagrin.
In both cases, the child is extremely disappointed when the parents attempt to replace the mechanical with the organic (in both cases an organic/living dog). Both the stories seem to bring up the question of what makes something alive/ethically valuable - is the robot dog's mechanically or digitally programmed loyalty any different from the organic dog's chemically and biologically programmed loyalty? And does it really matter in the eyes of the beholder - that is to say the eyes of the child?
Clearly it matters to the parents but is that only because they have been raised with the organic ideal and view the mechanical as unmistakeably "other". Differences between organic and mechanical beings are certainly greater than between different ethnicities but certainly one can see some sort of correspondence to a previous era in which other races were viewed as utterly "different" - as things to keep away from children.
The story inevitably brings up the idea of a soul - is the difference between robots and organic beings some indescribable phenomena or some unquantifiable or intangible essence? Asimov is (or at least as far as I can gather from his wikipedia page) an atheist and so I'm guessing he comes down on the side of the soulless - that is to say that he almost certainly thinks there is less of a difference between organic and mechanical beings than meets the ... well perhaps the eye in the earlier stories but when we move onto humaniform robots it will be the conscious.
But aside from the soul there seem to be two other elements that trigger a repulsion or a feeling of "otherness" associated with robots. First is their seeming mechanical nature - that is to say they respond to stimuli in entirely mechanical/determinant/fatalistic ways. They are machines are they not? Incapable of independent thought? These stories are certainly somewhat ambiguous in addressing this question - Asimov clearly gives Robbie non-determinant behaviors - that is to say emotions. And with the coming 3 laws in later stories the fatalistic nature of robots seems much more questionable as Asimov explores the weighing of those 3 seemingly simple laws.
The other element would have to be their uniformity - clones and the idea of cloning also seems to trigger the same problem. Anything that is uniform is to a certain degree replaceable - if I can produce thousands of identical robots are any of the individual robots special? To a certain degree the troublesome nature of this question seems to stem from the mechanical nature problem - that is to say we expect all robots of the same make and model to behave uniformly - which would in and of itself be proof of their fatalistic nature, proof of their un-uniqueness.
But what about twins? Certainly their uniformity does not take away from the special-ness. And they certainly are not mechanistic - nor are clones for that matter (for more on that see an interesting This American Life program on a clone of a famous bull - or was it Radiolab ...). Even the computer I'm writing this on has been shaped uniquely by the experience of working for me to such an extent that it is in many ways unique. Are robots any different? Do the sheer numbers produced - 20000 instead of 2 - make the difference? What if we could have identical vingt-uplets? Would the children seem so much less special?
A final note on the story relates to the thoughts of the parents in Robbie, worried about their daughter's lack of physical and emotional contact with other children and how that will stunt her development. The parent's worries seem to foreshadow future parent worries of the deleterious effects of television and then that most robotic of childish pursuits, video games. Having been raised at least partly on video games I hope the worries didn't prove to be too true.
So, I've analyzed these 2 short stories (albeit briefly) through the ideas of what it means to be human and alive, otherness and race and souls and clones and fate and video games. Perhaps lacking in the gender or economic department but all in all, a days work well done.