Margareta,
Your grandfather may have registered for the draft to gain citizenship, as shown below in the quotation from Wikipedia. The US Army was creating a division of Danish, Norwegian and Swedish speaking men to serve in Scandinavia due to the German occupation of Denmark and Norway. My father, who was a citizen, was drafted for this division because he was fluent in Swedish.
Conscription in the United States
From Wikipedia, the free encyclopedia:
Selective Service (and the draft) in the United States is not limited to citizens. Non-citizen males of appropriate age in the United States, who are permanent residents (holders of green cards), seasonal agricultural workers, refugees, parolees, asylees, and even illegal immigrants, are required to register with the Selective Service System.[41] Refusal to do so is grounds for denial of a future citizenship application. In addition, immigrants who seek to naturalize as citizens must, as part of the Oath of Citizenship, swear to the following:
... that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the armed forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law;[42]
Non-citizens who serve in the United States military enjoy several naturalization benefits which are unavailable to non-citizens who do not, such as a waiver of application fees.[43] Permanent resident aliens who die while serving in the U.S. Armed Forces may be naturalized posthumously, which may be beneficial to surviving family members.[44]
You can research his citizenship status through the website
http://www.jgsny.org/nynats.htm if he was naturalized in New York State.
John H.