This isn’t New York City, or the Windy City, or Sin City, and we’re certainly no one’s Emerald City'
What does "no one's city" stand for?
The question is not to understand what Emerald city is but what construction "no one's" is.
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"No one's" here is a way to say "is not" with extra emphasis.
For example, suppose someone asked you to deliver a message for him. If you don't want to do it, you might reply, "Sorry, I'm not a mailman." If you were offended by the request -- if you think you are too important a person to be asked to deliver messages or some such -- you might say, "Hey, I'm no one's mailman."
Sometimes it makes sense literally. Not only is something not true in this particular case, but it wouldn't be true for any person. In this example, if you said, "I don't think this is the Emerald City", a reader might understand this to mean that it isn't like the Emerald City for you, but it might be for someone else. But "this is no one's Emerald City" says it isn't like the Emerald City for anyone. (Of course it's still an opinion.)
But really it's an idiom and not meant literally. It means "very much not".
Note this idiom is normally used in the form "X is no one's Y", where X is a speicifc thing and Y is some class of things. BTW it's also often said as "is no man's" rather than "no one's".
Note that the same words could mean that it (whatever the "it" is you're talking about) doesn't belong to anyone. For example, if I read the sentence, "Bob is no one's auto mechanic" with no context, my first guess would be that the writer means that Bob knows very little about repairing cars. It's not just that he's not good at it: he's very bad at it. But in context it could also be literal. It may be that Bob knows a lot about repairing cars, and the writer is just saying that nevertheless he does not repair cars for anyone. Like most things in language that can be ambiguous, usually which meaning is intended is clear from context, though of course sometimes a careless writer may leave it unclear.