That sounds like the old Super Shuttle (which I know from CA, not NY).
I thought Uber's offering was more like a bus - you meet at the terminal and it takes you to the airport.
This is correct. They pick up at a small number of transit hubs and go direct to the airports.
They offer this at JFK and LGA but I heard the buses are empty, and their price is really low, so not sure it's going to work long-term.