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And for further social context, they are essentially saying what for a Dane is the equivalent of "New York is located in New Jersey".

No, it's more like "New york is located in Brooklyn". As in, Amager is still a part of Copenhagen, but does not encompass all of it.

Most of Amager isn’t part of Copenhagen. Dragør, Kastrup, etc, are their own municipalities.

I didn't touch on it, but there is largely a negative connotation with Amager. Its nickname is "lorteøen" which translates to "shit island."

Also, to be more accurate, it would be like saying New York City is located on Long Island.