This is a great question. For those who may not understand the reference to the "ATU folktale index" - a bit of context. The folktale type index was initially developed in the early twentieth century by the Finnish scholar Antti Aarne (1867–1925); it was then enlarged with a new edition by his American successor Stith Thompson (1885–1976). More recently (2011), a German folklorist, Hans-Jörg Uther has enlarged and updated the index.

Aarne's goal was to provide an international - but not worldwide - index for folktales, giving the folktales known to have been collected more or less from Ireland to India in what seemed to be a shared "bubble" of oral tradition. Some of the folktales appeared outside the bubble.

It is also important to point out that no nation could boast even a majority of these folktale types: Ireland proudly claims examples of over three hundred of the tale types, but in a study I did in 1982 of examples of literary sources at the Irish folklore archives, I was able to demonstrate that the number is actually slightly fewer than 300 - and yet, the catalogue is still extremely impressive.

As indicated, the intention was never for Aarne's initial index to apply to eastern Asia, Africa, Australia, Oceana, or the Americas. Folklorists in the early twentieth century hoped that Aarne's effort could inspire other regional indexes, and to a certain extent that has occurred - but not as much as hoped. Hiroko Ikeda, for example, published "A Type and Motif Index of Japanese Folk-Literature" in 1971.

Even as the Finnish Historic-Geographic Method advanced the idea of the type as a valid concept, it was assailed, particularly by the Soviet folklorist, Vladímir Propp (1895–1970). In 1928, he maintained that strict rules of composition provided the structure of the folktale, which artistic storytellers employed as they created new stories. To accomplish this, the narrator drew on tens of thousands of motifs, the elements of stories shared by everyone. These could be everything from Cinderella’s glass slipper to a ghost who is grateful for the burial of his remains. Propp argued that what appeared to be tale types was, in fact, an illusion caused by the repetition of traditional motifs constantly reordered into the structure of the tale.

When Propp's work was translated into English in 1958, it began to take the wind out of the Finnish sails just when other regions might have undertaken the arduous task of compiling indexes. This is particularly important for the way oral tradition has been studied in the Americas. In Europe, the study of human culture tended to fall into two camps with distinct bibliographies: ethnologists who studied the "colonies" and folklorists who studied their own, European folk cultures. In North America, departments of anthropology, which embraced ethnological methodology, engaged in the study of indigenous American cultures. Their methodology was not inclined to adopt the approach advanced by the Finnish and Scandinavian scholars.

A critical turn in the study of North American indigenous traditions from the point of view of folklorists was the work of Alan Dundes (1934–2005) who published his PhD dissertation, The Morphology of North American Indian Folktales in 1964. Dundes was one of the first Americans to employ Propp's approach to oral narratives. It appeared at a time when many American folklorists were drawn to the structuralism of Propp. By advancing Propp’s approach, Dundes was at the cutting edge of his field at the time, embracing the idea that narratives were inherently fluid, but it was not conducive to developing a North American tale type index. Indeed, Dundes argued that the American Southwest, for example, featured storytellers who continually changed narratives, making such an index difficult to compile.

In general, these efforts to find American narratives that fit the ATU index have not been successful. This should not come as a surprise since the index thrives on the idea of diffusion, and accounting for diffusion of these stories much beyond the Ireland-to-India zone was not expected to have been common.

The observations of Dundes in 1964 are to be contested, and I doubt that he would agree with all of his conclusions by the end of his long, enormously productive career as a scholar, but all of that - the nature of indigenous traditions and the way it was approached both by ethnographers and by post-WWII folklorists - inhibited the drafting of regional American indexes.

All that having been said, there have been attempts to study American Indian tale types: the Star Husband Tale, for example. And there have been attempts to see stories that recall the Old World Cinderella. More recently, there have been attempts to apply methods adapted from geneticists to identify tale types diffusion that includes the Americas, arguing a pre-migration inheritance shared by both hemispheres of the globe, but these studies are controversial.

In general, efforts to apply the ATU index to the Americas has not been successful. This is what would be expected since the premise of the method is to depend on diffusion to link similar narratives. One would not expect any given folktale to have commonly reached beyond the originally conceived Ireland-to-India zone.

I cannot speak to the final part of your question - about what American Indians have to say about any effort to categorize their stories. I will leave that to others.