Of course, the Dutch stole it from the Indians. While we are discussing myths and fairy tales, the one about buying Manhattan for 24 dollars worth of wampum deserves a dishonorable mention.
In addition to New Amsterdam, the city was New Orange for a time when ownership passed back and forth between the Dutch and the British. Way back then, NYC only referred to a little spit of land in lower Manhattan south of Wall Street. Wall Street got its name from the wall that stood there to keep out the invading hordes.