Speaking of examples, imagine a football club that has signed a sponsorship contract with Adidas, and then one player goes out on the field wearing Nike football boots.
You are wrong and this is happening all the time, but you are clearly not carefully watching football matches

Every player can choose any boots they like most and make individual deal, club sponsorship is only for shirts and kits.
You really don't need to embarrass him. examplens's example was not appropriate that's all but it was a good try. If the example was about an individual player sponsorship then it would be more appropriate however individual sponsorship also does not require you to wear their apparels when you are not in official tour or events.
In casual life you are free to use anything you like.