I lived in NY for many years -- including four years with a young child -- and have made many trips back that have included my children. It is true that many people do put young children, including infants, in cabs without car seats. But if you wouldn't consider driving a few blocks in your own town without one, I wouldn't consider it in the city, either. Speeds are often as high as they are on suburban streets, so even if it's legal, it isn't wise or safe.
I was in two accidents involving taxis in the 10 years I lived in the city, and an unrestrained child could have been injured in either one. When we've traveled, we've used a car seat to get to and from the airport, and have chosen to take the subway, ride the bus or walk everywhere once we got into the city. It can be done without much effort -- especially if you can travel with a sling, backpack or wearable carrier instead of a stroller. Our family is proof. In the four years we lived in Manhattan with a child, we did not put him in a taxi a single time, even though we traveled all over the city with him -- to the zoo, parks in Manhattan and Brooklyn, a pediatrician that was a significant distance away, museums all over the city, etc. It takes a bit longer to get there. But you save money, the kids usually enjoy the ride and the people watching, and you are far, far safer than you would be putting a child unrestrained into a cab.
New York is a great city to explore and enjoy with children. Good luck, and have a terrific, fun trip!