This is disclosed in https://www.uber.com/cities/new-york: "$20 surcharge on all trips between NYC and New Jersey." It doesn't have anything to do with where the driver is from. It also appears to be included in the fare calculator there. (I can't be 100% sure because it doesn't break down the bill.)

The fee shouldn't be dependent on where the driver happens to be from, or you'd be rolling the dice on which driver you happened to get: it's fairer to have a consistent price.