New York immediately makes me think of Manhattan. The island materializes so much the city core that it's difficult to not think about it apart. This being said, I also think later about the Bronx, the Queens and Brooklyn... but very rarely of Staten Island.
In my opinion, New York City is structured in a very similar way as Paris. And the funny thing is that I only see both cities which are structured as such. Both have a very well-defined city core limited by a clear border. For Manhattan, it is the fact it's an island in itself, and in the case of the city of Paris it comes from the périphérique surrounding it. Afterwards, both cities have an "inner belt" which are the three inner ring departments around the city of Paris and the three urban outer boroughs in NYC. And finally, there's an outer belt which is about all the outer suburbs sprawling all around both cities. When we think about, it's quite striking how this structure is similar.
We could pair them in that way :
City of Paris (75) => Manhattan
Hauts-de-Seine (92) => Brooklyn
Val-de-Marne (94) => The Queens
Seine-Saint-Denis (93) => The Bronx
The City of Paris is only a bit larger and more populated than Manhattan, but both areas have the same population density. The 3 inner ring departments in Paris are about the same size as the three outer boroughs of New York, but they are slightly less populated. And finally, the sprawl beyond is much larger in NYC than in Paris, even if it still represents nearly half of the population of Paris metro area.
In the end, there's NYC, a city of 18 million people, and Paris a city of 12 million people. So it's obvious NYC is bigger, and it feels bigger, I don't question this. Both cities are my favourite worldwide. Manhattan is still to me the ultimate urbanity. But Paris isn't that bad either when it goes about urbanity.