Yes. NY and NJ are fully independent and equal sovereign entities from one another. In general, most of the public road infrastructure is built and maintained by state and local governments (though there is some federal funding provided). Same with public transportation.

The main complication comes from the interaction between the states and the federal government.

This is a great example of how, even when one buys the idea of giving rights to individual states over the federal government, the US state borders are rather unnatural. Someone living in Jersey City and someone living in Manhattan have an incredible amount of shared economic interests. However, the Manhattan resident shares a state government with people living in Buffalo instead. This happens in plenty if places in the US: We have more than one multi-state megalopolis, and along with it, metro areas over a million residents that are split in such a way that they count little in their respective state governments.

In most of Europe, borders have had a whole lot of time to align economic development and political organization. But the US, unlike its corporations, is against reorgs. The fact that we even get to discuss state rights for something that could be a municipal matter is, in itself, a problem.