The New York Penny is a problematic term. First of all, it was called a duit, not a penny. Second, they were never made in New York. And, third, they were never even meant to be in New York.
When the Dutch Republic founded New Amsterdam on Manhattan Island in the early 1600s, they brought their money with them. The smallest of these coins was the copper duit.
The English captured the settlement in 1665, and the Dutch formally ceded it in 1667. The English then renamed it New York. Despite this change of ownership, Dutch duits continued to be in circulation (as were other coins, particularly Spanish ones) throughout the colonial period.
The VOC Duit
Ironically, the duit specifically marketed today as the New York Penny was never meant to go to Colonial America. These coins, marked with a prominent VOC, were minted by the Dutch East India Company, or Vereenigde Oost-Indische Compagnie, abbreviated to VOC. The VOC operated in the area of Indonesia. However, their coins were in wide circulation and did reach the Americas.
To be clear, the VOC duit was not minted by the Dutch government. The VOC was an immensely powerful company, and the government granted them the right to mint their own money for use in their colonies. This VOC duit bears its initials on one side, along with the date, and the crest of the province of Holland, which is where this particular coin was made.