Different writers come up with the same ideas all the time -- in fact, we generally try to avoid copying other people's ideas but often end up accidentally doing so anyway. There are only so many ways to tell a story, and we're all working in the same cultural zeitgeist and drawing on the same conceptual vocabulary, so inevitably different creators will come up with similar ideas without specifically trying to. So similarity alone is not evidence of direct inspiration or imitation. Indeed, it's usually evidence that a writer wasn't aware of the other work, since if they'd been aware of the similarity, they would've changed their idea to make it more distinct. Nobody wants to be considered imitative, and nobody wants to be sued for plagiarism.
In this case, it's easy to see where the inspiration for both stories came from. A lot of stories come from role reversal, challenging readers/viewers to imagine themselves in the reverse of their normal situation. What if humans were the ones in a zoo rather than animals? What if humans were the ones being hunted, as in The Most Dangerous Game? What if humans were the food animals, as in "To Serve Man"? The common theme is to encourage empathy, to make us see things from the perspective of the creatures we exploit and ask ourselves if we have the right to treat them that way. You can find plenty of other role reversals in science fiction, like, what if women were in charge and treated men like second-class citizens (as in Roddenberry's Planet Earth pilot and TNG: "Angel One")? One of the unused story seeds (fortunately) in Roddenberry's original pitch document for ST was about an alien world where black people enslaved white people. Clearly Roddenberry was interested in stories that put the characters in reversed or alien roles.
Indeed, that's one of the most fundamental themes of science fiction -- imagining the Other. Positing something different from everyday reality, encouraging the reader or viewer to see things from an alien point of view or imagine a life radically different from the one they know. And one of the most fundamental human drives is empathy -- looking at another being and trying to imagine how they experience life. People have probably been coming up with "what if humans were the ones in the zoo?" stories since the very first zoo was established. (Although, for a long time, humans did put other humans in cages and treat them like animals. Some still do. So many people didn't have to imagine that.)
Anyway, I think "People Are Alike All Over" and "The Cage" are different enough that Roddenberry wouldn't have been worried about the similarity even if he had been aware of the earlier episode. "People" is about an astronaut believing he's found paradise on Mars and only learning at the end that he's an animal in a cage. "The Cage" is about a man aware from the start that he's a prisoner and being offered the illusion of paradise as a temptation to submit. Character-wise, "People" is about a man's optimistic hopes being dashed by a hard reality, and "The Cage" is about a tired, cynical man gaining new hope and resolve through his ordeal. The same concepts and devices show up throughout fiction, but different writers can do very different things with them, just like Rachmaninoff and Little Richard could do very different things with a piano.