I disagree that this is an existing thing in sports.
Not at all in the NFL.
Not at all in NHL.
Not at all in any college sport.
Minimally in MLB, but only because the Dodgers left Brooklyn, and Yankee fans (ie more than half of NYC) don't much care. The media would go crazy over a new Yankee/Dodgers World Series. I'm not sure average fans in either city would care at this point.
Minimally in the NBA, but based on little more than a single pair of championship series that are now 45-50 years gone. Since then, there is buzz in NY when the Lakers come to town but I'm not sure that has been consistently true in reverse. Due to quality Boston/LA has had more sustained heat than NY/LA since the 1950s.
This doesn't mean it can't work in MLS, but there's no real sports predecessor.
Outside of sports I agree there is a NY/LA rivalry in the larger culture. I think it is media driven -- and fascinating more to people within each industry rather than to the general public -- but I acknowledge it is real. Also, I suspect most people in between will yawn and tune out. They care much less about NY and LA than most people in those bubbles realize. The ones who care the most move to one of those cities. The ones who stay find the NY/LA stuff extremely tiresome.