The recent U.S. move to blacklist Huawei Technologies Co. mainly targets the Chinese giant’s market-leading telecommunications-equipment business. But Huawei is also one of the world’s biggest smartphone makers, and the U.S. export ban now threatens to cut off its access to crucial phone components and software for devices used by millions of people world-wide.

The U.S. sees Huawei as a national security threat, and most Americans have never used a Huawei phone. For many people around the globe, however, the brand is known...