You almost have to think someone is literally making this shit up but lo’ and behold it’s true.
Some non-citizens are required to register. Others are not. Noncitizens who are not required to register with Selective Service include men who are in the U.S. on student or visitor visas, and men who are part of a diplomatic or trade mission and their families. Almost all other male noncitizens are required to register, including illegal aliens, legal permanent residents, and refugees. The general rule is that if a male noncitizen takes up residency in the U.S. before his 26th birthday, he must register with Selective Service. For a more detailed list of which non-citizens must register
Bold mine from the Selective Service themselves