Yes. Most of New York State runs APS county-by-county through each county's own Department of Social Services. New York City -- including Richmond County/Staten Island -- centralizes APS under the NYC Human Resources Administration (HRA), Office of Special Services, with borough-specific phone lines rather than five separate county-run agencies.
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This distinction matters if a family is comparing information found for a different part of New York State, since guidance written for an upstate county's APS structure will not directly apply to how reporting works on Staten Island. The practical effect for a Staten Island family is a single centralized system serving the whole city, reached through a borough-specific line for the initial report.
This structure also means Staten Island's APS resources are shared citywide in some respects -- policy, training, and administrative oversight run through one NYC HRA structure, even though the borough-level phone contact is Staten Island-specific.