There is more than one "New York." The name "New York" can refer to the State (which was named for the City, and not vice-versa), the City (the official name of which is "New York" not "New York City" or "The City of New York"), and the County of New York, better known as the Borough of Manhattan.
There is a West New York (in New Jersey) and the neighborhood of East New York (in Brooklyn).
But did you know that Seattle, Washington was originally named "New York," when it was settled in 1851? In 1853, the name was changed to "New York Alki" (a Native American word meaning "by-and-by"), and a few months later to "Seattle," after the log-cabin settlers realized they just were never going to be able to compete.