To my understanding it mostly depends on the profile of the sport and the school. If Zion Williamson got on the court with some Adidas basketball shoes then I'm sure Duke gets a call from Nike to the effect of 'take off the Adidas shoes or we will stop sending you free stuff.' By this standard most every track team in the country is flying below the radar of the big shoe companies because they aren't racing on national TV 9.9 times out of 10. Notable exceptions might be NAU, BYU, Oregon, Colorado, Arkansas, etc.
I ran D1 at a crappy school with an Under Armor sponsorship. Some years the department would tell us they couldn't buy anything but UA (try using their distance shoes I dare you) and some years they would tell us they just couldn't buy a big competitors shoes (Nike, Adidas, Brooks). One time I got Mizuno shoes for a season. One year we bought Nike shorts and put our school logo over the Nike logo since they weren't UA. One year our 4x4 team won conference and the coaches had them take off their Nike spikes and socks before the picture on the podium.
Basically, I'm saying if you're a small school who mostly sucks then you can probably wear whatever you want. I trained and raced in either Nike or Adidas all four years. If you're a big money school playing on TV then you probably have to be a lot more careful about what you wear. That's why (I assume) Nico took the Nike logo off of his Dragonflys in the 10k a couple weeks back.