Because New Amsterdam was a colony of the Dutch West India Company, many of the first property owners obtained their land through a Land Grant or Land Patent. The owner is given the Land Grant or Land Patent typically by a legal entity that claimed the land.
Examples:
Dutch West India Company divided the land on Manahattan Island, and sold lots for a fee or granted them outright as Land Grants.
The term "Land Grant" is often used for any transfer of land, including the first. The term "Land Patent" is only used for the first owner to originate title. It is similar to what we usually think of as a patent, where someone is granted their claim that they were the first to create a product or idea.