Unfortunately, the answer here is that the traditional Nicolet story is a myth. A very thorough assessment of Nicolet's expedition (and context) was published in 2018 (shorter synopsis here) and the underlying research provides a more grounded (but still interesting) take on what he was actually doing (and wearing). To be sure, Nicolet's superiors in French North America had once entertained fancies that the western Great Lakes opened into the Pacific (and to China beyond it), and had expressed mild interest in tales of a distinctive Native American people west of the lakes (likely the Mandan). But they understood by 1634 that they were nowhere in the immediate vicinity of China.

As for Nicolet, during his 1634 voyage, he was acting as a diplomatic emissary of the Company of One Hundred Associates, which at the time held a monopoly on the Quebec-based French fur trade in North America. Nicolet was the definition of a "company man," having previously working as a clerk, interpreter, and resident fur trader in Algonquin communities in modern Ontario. At the time of his 1634 landing, he was acting as an agent of the company's leader Samuel de Champlain, who wanted to make contact with the Ho Chunk people of modern Wisconsin. The Ho Chunk were engaged in warfare that Champlain wished to curb, and Nicolet's task was to try and forge a peace treaty.

When Nicolet first landed (perhaps near modern Marinette, Wisconsin, not Green Bay, as often perceived), he wore, according to contemporary accounts, an outer garment made of "Chinese damask." In all likelihood, this simply referred to an elegant silken garment in vogue at the time among people of prominence in Western Europe and their North American settlements. As the diplomatic representative of France's economic machine in North America, it would make sense that Nicolet would dress in such a garment (perhaps the finest he owned?) when meeting the Ho Chunk for the first time.

The notion that Nicolet was dressed in Chinese formalwear, and under the mistaken impression that he was landing in China, seemingly descends from a misunderstanding by later historians translating sparse contemporary accounts of Nicolet's expedition. It seems that a reference to the aforementioned Chinese damask robe was misconstrued as a reference to an explicitly Chinese-style robe, with more fanciful elements added in over time.