It's less that the state is named after the city or vice versa, and more that both geographical regions were given Anglicized names at the same time. While the legacy of Native tribes and people can be seen in place names such as Manhattan (based on a Wampanoag or Lenape word used to describe the land mass between the Hudson and East Rivers), name places in New York State reflect the region's appeal to a variety of European colonizers.

Prior to the arrival of Europeans, a collection of Indigenous settlements and communities were spread across what would become New York State, parts of Pennsylvania, and New Jersey. The region, described by many Europeans as "Iroquoia", would become the home of the Iroquois League of the Five Nations (Mohawk, Oneida, Onondaga, Cayuga, and Seneca) sometime between 1450 and 1550. Eventually, a sixth tribe, the Tuscaroras, joined what would become the Haudenosaunee Confederacy. Members of the Confederacy traded with Europeans on an individual, village, and nation basis and multiple name places reflect their presence. Counties in Central New York carry the names of the tribes of the Confederacy. Counties in Western New York (e.g. Erie, Chautauqua, Cattaraugus) take their names from tribes from the western part of the state or place-names used by members of the tribes.

Dutch colonizers from the Dutch West India Company lay claim to the land inside the Confederacy, on Esopus land along the Hudson, and on Wampanoag, Munsees, or Montauk lands on the state's southern islands and identified it as New Netherlands. In doing so, they basically ignored the early mapping activities of Italian Giovanni da Verrazzano (more on colonial naming practices from /u/sunagainstgold). They identified the region at the bottom of the colony as New Amsterdam and a region about 150 miles north along the Hudson, as Beverwijck.

England took over the land 1664 and immediately renamed several key locations.^1 The colony itself and what had become the two largest settlements in the colony were named after the Duke of York and of Albany, who, in 1685, would become James II of England. The settlement on the Hudson would be chartered as The City of Albany in 1686, making it the longest continuously chartered city in America. New Netherlands became "New York" and the area that was New Amsterdam also became "New York." The City of Greater New York was established in 1898, bringing together the various boroughs into the city we recognize today as The Big Apple. New York County, which originally included parts of the Bronx and Staten Island, now includes only the NYC borough of Manhattan.

But to be sure, the problem was noted from the very beginning. One author in mid-1690's noted that England had "planted seeds of confusion across the path of one who would seek the meaning of a New Yorker. No other state has to deal with such confusion.”^2

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1. For more on the transition from Dutch to British rule, this article from 1901 gives a pretty detailed timeline of events.

2. Reitano, J. (2015).New York State: Peoples, Places, and Priorities: a Concise History with Sources. Routledge.