It is a common misconception that Staten Island and Manhattan are standalone islands separate from New York City. In reality, both are integral boroughs of New York City, and they are part of the same archipelago that forms the city's geography.
New York City is built on a collection of islands at the mouth of the Hudson River:
So, Staten Island and Manhattan are indeed islands, but they are not "their own" in a political or administrative sense—they are components of the City of New York.