Geographically, Brooklyn, as well as Queens–both boroughs of New York City–are part of Long Island, along with Nassau and Suffolk counties.
Though most Brooklyn or Queens residents would strongly reject the notion that they live on Strong Island, it’s true! Just look at the map.
Manhattan and the Bronx (and Staten Island, but no one really cares about my forgotten borough–hell, it’s not even important enough to have a native SCOTUS justice) are distinctly separate from the Long Island that Brooklyn and Queens call home.
So the New York Islanders may be moving to Barclay Center in Brooklyn, but they aren’t leaving Long Island. And the “Islanders” name still works!