This question might be better asked at r/askliterarystudies. Still, I'll add what I know about it.

Castaway narratives are not new. The castaway has been an enduring theme throughout history, as a glance at various Wikipedia entries on the topic will show.

Still, how do Enlightenment castaway narratives differ?

It is important to note that such narratives were concomitant with other important historical events, such as the creation of colonies by European countries other than Spain or Portugal. It was concomitant also with an expansion in travel for those who were not part of the elite at the end of the 18th century and into the 19th. This was also a time when all of Latin America was suddenly open to foreign travelers, after Alexander von Humboldt "rediscovered" that continent and was hailed as the "second Columbus." This was also after the new republics got rid of Spain's policy of keeping foreigners out; this inaugurated a "re-invention" of Latin America, to use Marie Louise Pratt's term.

Given this situation, it is hardly surprising that new colonial enterprises would enjoy fictional accounts of the lone European stranded amongst the savages, who yet asserts his or her superior will and educates and teaches the natives, so as to be able to claim, as Robinson Crusoe did, that they were kings in these new realms.

If we take the case of Germany, Susanne Zantop argued that because Germany was not a colonial power until the very end of the 19th century, they created "fantasies" of German colonies in literature. Castaway narratives showed how Germans would be better colonists, if not the best. This is especially important to note since the new colonial powers were not "backwards" and "unenlightened" Catholic nations, like Spain and Portugal. Spain and Portugal did it all wrong. The enlightened Protestant nations of Europe would show how to do it right, in the moral sense or in the economical sense.

Pratt, Marie Luise. Imperial Eyes: Travel Writing and Transculturation.

Zantop, Susanne. Colonial Fantasies: Conquest, Family, and Nation in Precolonial Germany, 1770-1870.

EDIT: fixed typos and clarity