In NJ do registration and insurance ID card addresses have to match

Yes the addresses need to match in NJ.

Generally the address listed on your registration must match the address on your insurance if you are the primary owner of the vehicle. This is true in pretty much every state including New Jersey. The reason for this boils down to practicality. Your physical address is required of both registration and insurance. In regards to registration this has to do with identity, but for insurance this has to do with where your car lives. This rule pertains only to the actual owner of the vehicle, and is not necessarily mandated for other drivers.

Car insurance companies have to know where the car will be permanently parked. This means the physical address of where you live has to be listed. You can provide a separate address if you happen to receive your mail at a different location. When applying for registration at a state DMV, your physical address is one of six forms of identification you have to provide. Although you can list more than one address here as well, since many people receive mail at a different location, your home address will also be on file. So if a police officer was running your credentials, your home address would appear on both.

In many states this is a mandate, but in others it is recommended but not required. If you change your address and do not update it with the DMV, generally this is not a major problem. You may have to answer some additional questions when pulled over, but most of the time you will be fine. However, some insurance policies do not provide full coverage unless both addresses match. So in any case, it is wise to check with both the DMV and your insurance company to see exactly what each requires. Your local DMV will be able to corroborate this information.