"Amaterasu" certainly seems to refer to the Shinto goddess, but the full name for the deity is Amaterasu Omikami (天照大神). This name was not trademarked, as it's unlikely that the Japanese Patent Office would allow Nintendo to copyright an actual god or goddess.
Amaterasu Omikami is a sun goddess from which the Japanese imperial family claim a direct lineage. During the ancient "Jomon" period in 660 B.C., the country's first emperor Jimmu claims to have descended from Amaterasu Omikami and founded the Japanese empire. The Japanese imperial calendar, which stopped being used after 1945, commences on the first year of Jimmu's reign. But the story is myth, the stuff of legends. And during World War II, it was the stuff of propaganda.
Since the late 19th century, the country's war flag depicted a rising sun — something that came to mean Japanese expansion as WWII ground on. Kamikaze pilots associated themselves with Amaterasu, while the idea that the emperor (and even the Japanese people) were descended from the heavens was used to explain away the country's mid-century land and natural resource grab. All the "divine" talk peppered in Japanese was ultimately cover for military aggression.
Amaterasu Omikami is still very much a fixture in Japanese traditional culture as well as popular culture. Video games such as Okami and Shin Megami Tensei: Persona 4 feature the deity. While Nintendo's trademarked "Amaterasu", don't expect the Kyoto-based studio to use its newly found trademark to explore 20th century imperialism.
Culture Smash is a daily dose of things topical, interesting and sometimes even awesome — game related and beyond.