Nobody owns 'Who Dat', which is as it should be. The history of Who Dat starts with it being a minstrel and vaudeville show phrase from the late 19th centuries, with mutations into stage and screen, stop offs as boredom relievers for american soldiers and airmen and most recently team chants for New Orleans sports teams.
Between 2010 and 2012, the NFL tried to claim ownership of the phrase as a trademark of the Saints but was beaten back by a broad coalition. The phrase is now firmly in the commons once more, which means that it's available for all to profit from. This is as it should be. I don't begrudge the NFL making a bit of money off of Saints merchandise. The NFL likewise shouldn't be offended at others who do the same thing. The right to copy is fundamental.