Why would NJ resident file a NYC return? NYS yes, and income tax credit and COJ for NYS tax.

There is a NYC filing (NYC UBT)_ and tax for unincorporated business operating in NYC. A NYS tax return is also filed.

Is the taxpayer a NJ full year resident? 

I think this would fall under credit for tax paid to another state/locality.

Unless NJ cooked up something special for self-employeds.

See Example 12 on page 40 https://www.nj.gov/treasury/taxation/pdf/pubs/tgi-ee/git3b.pdf

The first problem is that the tax prep software does not fully handle this. It is partially handled with only one NJ-COJ which does not give the client the full credit for taxes paid to another state/city etc. Each additional part of this requires an additional NJ-COJ which the software does not handle or prepare. The additional forms and calculations need to be done manually.

Of course, I did read the instructions which I found to be as clear as mud.

I then reached out to the NJ tax preparers helpline and was pleasantly surprised to get through to someone in a very reasonable amount of time. I was also surprised that I reached an agent who was knowledgeable, helpful and pleasant. Even so, she agreed that the instructions were at best confusing and certainly not very clear. After some more research by her and her speaking to some of her colleagues she helped to give me an important piece of the puzzle and with some additional research on my part I was able to get to the solution. 

Interestingly in speaking to a few other CPAs in the NY/NJ area there didn't seem to be much knowledge or information about this. It seems that many if not most preparers fully rely on the program and don't follow through on the points that fall through the cracks.

I would agree with sjrcpa  that; 

It's amazing what you can find when you read the instructions.

Yes.

Correct.