I rather disagree with Werrf’s answer. Asimov wrote Robbie in 1939 when he was only 19 years old. He was still quite inexperienced then, and I think he simply wrote a straightforward story without any metaphors in mind. As mentioned by Broklynite, there were two types of robot stories at that time. Mostly, robots were portrayed as a menace, and Asimov couldn’t stand those stories. Then he came across two stories which featured sympathetic robots, and they inspired him to write a story himself about a robot that is good and lovable.
I think that’s all there is to it. Robbie is a story about a sympathetic robot, and insinuations about man’s dependency on technology etc. were not intended by Asimov.
[I]t became very common, in the 1920s and 1930s, to picture robots as dangerous devices that invariably destroyed their creators. The moral was pointed out over and over again that “there are some things Man was not meant to know.” […].
At any rate, without quite knowing what dissatisfied me about the robot stories I read, I waited for something better, and I found it in the December 1938 issue of Astounding Science Fiction. That issue contained “Helen O’Loy” by Lester del Rey, a story in which a robot was portrayed sympathetically. […]
At almost the same time, in the January 1939 issue of Amazing Stories, Eando Binder portrayed a sympathetic robot in I, Robot. This was much the poorer story of the two, but again I vibrated. Dimly, I began to feel that I wanted to write a story in which a robot would be portrayed lovingly. And on May 10, 1939, I began such a story. The job took me two weeks, for in those days it took me quite a while to write a story.
— The Story Behind the Robot Novels (1983)
Regarding the Three Laws, DJClayworth and Werrf are right of course: They didn't exist yet when Robbie was written. The Three Laws of Robotics were first featured in Runaround, which Asimov wrote in 1941. But incidentally, Robbie contained a sentence that Asimov later called a “first hint of the First Law”: “He just can’t help being
faithful and loving and kind. He’s a machine—made so.”