First the media misquote Donald Trump. He does not want to register all Muslims, he simply wants a registry of any Syrian refugees we let in.
Second, the media is ignorant of the law. From U.S. Immigration:
What is an Alien Registration Card?
February 1, 2011
An alien registration card in the United States is also known as Form I-551, a green card, or a permanent residence card. This card gives the holder a way of proving their legal status to live and work permanently within the United States. It is important to note that permanent residents are required by law to keep their alien registration card on their person at all times. Failure to do so can result in penalties, a criminal record, and fines. It is also illegal to carry an expired, inaccurate, or damaged alien registration card. Your card must be kept in good condition, kept up-to-date, and kept on your person at all times.
There are many reasons why the United States requires a green card for permanent residents. In the US, a green card gives residents who are eligible the right to live and work permanently in the United States. Green card holders can travel freely within the United States, can enter and leave the country more easily, and can hold down jobs with virtually any employer within the United States. The green card is also the first step to US citizenship or naturalization. Anyone who wishes to go through the naturalization process in order to become a US citizen must generally secure a green card first.
For the US, the alien registration card provides a simple way to separate undocumented and documented residents. While the United States welcomes immigrants from other countries who can help build the nation and can help contribute to the US economy and to US culture, undocumented workers and undocumented residents are a serious threat to public security.
An alien registration card can help authorities distinguish who is legally allowed to live and work in the United States and who is not. For employers, an alien registration card can help them determine who they can hire and who they can not. If a potential employee cannot produce an alien registration card, for example, the employer may have a simple way of determining whether that person is authorized to work in the United States. For law enforcement and border security professionals, the card allows them to determine a person’s eligibility to enter or reside in the United States. Permanent resident status is a highly prized status and those who have it can always easily prove their status within the United States by carrying an up-to-date green card.This is how it should be.