Supreme, an underground streetwear brand, has attracted a cultlike following among teens seeking hard-to-get looks by keeping its products scarce. But now an Italian businessman is flooding the market with T-shirts and hoodies bearing its bright red-and-white logo, and claiming he is doing nothing wrong.

Michele di Pierro, who started the rogue version of the brand in Europe, has filed trademarks for variations of the Supreme logo in dozens of countries since 2015, forcing the U.S. company to face off against him in courts...