No—Manhattan is not the capital of New York State. This common misconception stems from Manhattan's outsized cultural and economic prominence as the core of New York City.
Manhattan houses Wall Street, the UN Headquarters, Broadway, Central Park, and iconic landmarks that dominate global media representation of "New York." Its 1.6 million residents (on 22.8 square miles) generate disproportionate economic, cultural, and political influence. But administratively, the Governor's office, State Legislature, and highest courts operate from Albany's Empire State Plaza complex.
Other states with similar "largest city ≠ capital" dynamics include California (LA vs. Sacramento), Texas (Houston vs. Austin), Florida (Miami vs. Tallahassee), and Illinois (Chicago vs. Springfield).