As that same Wikipedia article says, the US generally pays no attention to Frobisher.
The previously mentioned Wikipedia contains some much-needed clarification on the American “Thanksgiving.” To amplify just a bit: the Plymouth “Pilgrims” (they did not refer to themselves as such - they were “The Saints” and those among them who were not of their exact religious persuasion were “The Strangers”) were, technically, Separatists, who advocated total separation from the English Church, as opposed to the Boston Puritans who, as the name implies, were in favor of purifying the English Church of any remnants of the Roman Church. The Separatists had virtually no religious ceremonies; even marriage was considered a civil rather than religious function. Their concept of a “thanksgiving” was to thank God for an extraordinary intervention in human affairs. Although we might justifiably think that survival after the first winter in New England called for some serious giving of thanks, it apparently did not meet their established criteria. It was a harvest festival, not a religious ceremony, although certainly some typically restrained prayers were offered. Another misconception: the primary motivation for the “Pilgrims” coming to New England was not the quest for religious freedom. The Dutch Church was completely accepting of their beliefs and practices, but being Englishmen, they were essentially second-class citizens in Holland, generally unable to practice their chosen professions, and they feared that their children were growing up more Dutch than English. They, and some co-religionists who still resided in England but were very much a minority, were enticed by a commericial enterprise to emigrate and establish a (hopefully) financially profitable colony where they could continue to be dutiful subjects of the Monarchy while espousing their own religious beliefs and practices.