Well it shouldn't be classified as a NY game. Infact they really shouldn't be referred to as New York at all which most likely would antagonize their fans more than anything. Obviously its the nearest MLS club (at the moment) but some people are treating RB like they are guaranteed to be the only rival. There are exceptions of course. In the case of MCFC for example one rival stands out far above the rest but Man United are such an utterly detestable club so contempt is almost entirely directed at them.
You never know what could happen in the first couple of years with DC, Philadelphia, NER, etc. There might be a game or incident of such magnitude that propels it right to the very top of the list. Look at Yankees for instance they don't consider the NY mets to be their main rival. Right now its too early to know for sure though it probably will be RB.
Speaking of RB there's one key issue people overlook with them. RB is a corporation and corporate policy can change at any moment. They might not even be called RB 10 years from now. Obviously someone would buy it but I would suggest their ownership is far more uncertain than that of NYCFC or Cosmos. When the usual tedious argument is directed at NYCFC by their fans that really will be one of the main ways to fire back at them. At least this club knows the owners are in it for the long haul.