Trademark. Not copyright. Nike is the name of an ancient Greek goddess, but try marketing some kind of sports equipment named after her and see how far you get.
Trademarks are limited in scope - that is, they only cover the situation when the phrase is used to market or sell certain things (and those "things" have to be explicitly listed, by category, in the trademark application or notification). So you might get away with marketing avocados, or ball bearings, using 'Nike'. But not shoes or bags.
In this case, it looks like the publishers have dibs on clothing bearing the phrase, and the website allows you to print it on T-shirts. Hence, infringement.
Not mentioned in the story is whether the website would, in fact, allow you to print this image on a T-shirt. If it would, then much as I hate to admit it, the sharks were right. If not, then I'd say the family should have a good case.