There's a few ways to answer this. There were rivalries between all the sky-scrapers being built during the period and whoever had the tallest building was going to win. There was a small rivalry between John J. Raskob and Walter Chrysler.

The Chrysler Building secretly added their spire as a last ditch attempt to beat out the Bank of Manhattan building that was being built simultaneously.

Beyond that, everyone had a rivalry going, but these days we love our neighbor the Chrysler Building.