Former Poll Worker here.

New Jersey doesn't require ID to vote, however, there are a few exceptions.

When you first register, unless you never did it at the DMV, you will be asked to give your license. You can use your social if you don't have that (or a state ID). We do have a few individuals on our books that will mention they need to be verified, whether that's to make sure their address is correct (especially for first-time voters) or some other rule. I don't know why else they do it, but we may ask to just have a form of ID if it is explicit. Otherwise, we will not ask, we are discouraged from asking.

For Proof of ID, both photo-id and non-photo ID, including the sample ballot, a bank statement, even your Costco card, works.

When we switched from a yearly to a permanent vote-by-mail system, my mail-in ballot never arrived and I had to vote with a provisional ballot.