This piece, During Rule 40 Blackout, Emma Coburn Showcases New Balance on Olympic Stage, FloTrack, has a good run down of Rule 40 and how Emma Coburn cleverly circumvented it. In short, Rule 40 prevents all but official brands and whom they sponsor from promoting them during an blackout period of time surrounding the time of the Olympics. For example, US athletes using Nike can promote the Nike brand, but US athletes using other brands like New Balance cannot.
How did Coburn circumvent this? According to that article,
After crossing the finish line in third behind Ruth Jebet and Hyvin Jepkemoi, respectively, Coburn immediately removed her New Balance spikes and draped them over her shoulder before carrying the American flag. As a result of the bold move, thousands of photos snapped during her victory lap included her sponsor, New Balance, which otherwise would not have been featured. It’s more than likely that Coburn, who is vocal about sponsorship rights, did this intentionally to spotlight New Balance in the middle of the Rule 40 “blackout period” and circumvent Nike’s exclusive sponsorship rights with USATF.
One thing to note is that there are different rules for different athletic federations, it seems. The US swim team has more latitude than the track and field athletes.
As always, this is about money. Whatever else the Olympics are about — and obviously they are about many good things — money is one of the big aspects of these games.