Asking about the differences between oral histories of all the cultural peoples of multiple continents is a pretty vast question, but I think something can still be said about it. First, instead of trying to do a comparison between Europe and various cultures of the Americas, I'm going to look at purposes of oral cultures, and how these functions might impact how people pass on history - and I'll try and give examples as I go from different cultures. Secondly, I'll look at some factors that can negatively impact the passing on of oral histories and stories.

There are a number of reasons for people to pass on stories. Primarily stories are a mentally efficient means of connecting events and meaning, and most oral cultures have a body of stories that to some degree encompass a meaningful part of what is considered to be an education. All Jewish kids were expected to learn the Torah, Christian kids are expected to learn some additional stories, Kazakh kids are expected to engage with yet a different body of literature, as are Arab kids, Nuxalk kids, Gwich'en kids, Cree kids, and so were Icelandic children, ancient Greek children, Roman children, and so on. That's not to say that trades and skills were/are not an important component of an education in these and other cultures, just that when it comes to learning how to think, stories are the thing. So all of these cultures took a certain level of care with the passing on of their oral histories.

Since these stories are supposed to engage thinking to some degree, storytellers learn how to draw out ideas, decisions, challenges, and so on for their listeners to engage with. The stories aren't necessarily to be exemplary (take trickster characters or the acts of Greek and Roman deities), but they are supposed to elicit a response.

As an education tool, many of these stories also have a certain just-so quality, explaining how things came to be, and giving a framework for further engagement with the physical world.

At times the accuracy of the stories becomes very important, for example the large body of oral histories that have as one of their function the sharing of geographical knowledge such as many Pacific Northwest oral histories, or many of the Viking sagas as developed in Iceland, or in the Songlines of Australia. These stories are grounded in a landscape, and as such have some very fixed structures, at least as long as people remember the purpose of the stories.

Where the details come into their own is when these stories become not just explanatory in nature but also take on the role of creating social realities - when these stories become legal documents. Some of the oldest written texts we have are of law codes, as well as of the stories that gave the creators of these codes the power to create them. Do you want to found a city state where you have absolute power? create a story that says you are descended from a god. All of a sudden the details become important in a different way - rather than just making sure the proper way of thinking is passed on (through a shifting collection of relevant stories that have similar themes and lessons), it becomes important to pass on very specific details about who is who's child, and so on.

Here's a case for accuracy - in many First Nations cultures, everything from names to land to position are passed down connected to stories which are a part of oral culture/oral history. Because everything is tied in to them, accuracy is really important, and things get passed on.

So in summary, we pass on oral histories/stories for a variety of reasons, which impact the accuracy.

I answered a question regarding North American indigenous oral literature here - in my answer I focused on the factors that lead to the break down of oral history, as well as things that motivate the passing on of important historical narratives. I would encourage reading it, but the summary is that traumatic events have a profound impact on oral culture. Entirely new oral storytelling cultures can build up as a way of helping a culture or community work through and emotionally/mentally survive traumatic experiences, and these traumatic experiences can interupt oral transmission through death, or through a variety of other disruptive mechanisms such as physical displacement or linguistic displacement, all of which reduce the relevance of preceding oral histories. In my response I outlined how my own Metis Nation has definitely experienced shifts in focus with our oral storytelling practices, which has let very important events to be glossed over as we try to come to grips with the massive impacts of colonization.

When we start looking at these types of traumatic events in the Americas and in Europe, we find that there are patterns in history that have resulted in different approaches to oral histories. First - place. Europe over the past three or four thousand years has not been what I would call a stable community. Two thousand years ago, France, Spain, most of Italy and Romania were not Romance speaking areas. England did not speak a Germanic language, and in fact Scotland did not even speak Gaelic! Egypt to Morocco did not speak Arabic, and Turkey spoke Greek (as well as other languages). In other words, there have been some pretty serious changes. Migrations, wars, conquests, brutal displacements of the populations of much of europe mean that very little of the earlier world's oral history remained relevant. If Hebrew scriptures survived, it was likely at least in part because of the extent to which the culture around them was focused on how to deal with traumatic experiences and rebuild after them (think the Exodus, Babylonian captivity, Ezra and Nehemiah, and so on). Every physical displacement meant that most just-so stories were no longer meaningful. Every linguistic and religious displacement meant that oral narratives to teach critical thinking had become obsolete with the availability of a new set. Some of the oral histories that are renown for having survived would be the Viking sagas, which were written down just as they started to loose relevancy, and then remained important because Iceland was largely cut off for half a millenium. Another would be the Greek stories of the Odyssey and the Iliad, which survived because they were written down and became textbooks for the learning of classical languages. French and German folktales survived thanks to the Grimm brothers and others, and through being free public domain sources for television plots.

In contrast to this history, many cultures of the Americas have a very different background. Many nations have been in the same exact areas for several thousand years, sometimes ten or more. Those who have moved may have still been in the same areas for a thousand years or more. While seriously traumatic events have taken place, many of these events took place recently enough that communities still recorded the previous millenia-encompassing histories for their posterity.

That said, there are still many cultures that have been more mobile, and if you for example look at Cree oral history, much of it is not place specific, nor focused on the "history" part, but instead is a lot of creation stories that are related to maintaining an effective lifestyle. If you look at the oral histories of the far north, those of the Na-Dene related peoples often do have stories going into the deep past, and this is at least somewhat helped by the fact that they still live in the same environment - and if you look at their Navajo and Apache relatives, many of these stories seem to have been displaced by Hopi and other myths that are more closely connected to their current locations and lifestyles.

So to summarize a probably overly long ramble - there are a lot of reasons to maintain and pass on oral histories (and I didn't even get in to means of passing them on and maintaining them), and there are a lot of things that make this difficult. Europe and the americas have been impacted by these factors, but it's definitely true that europe has had a lot more cultural shifts, displacements and traumatic events over the past few thousand years, which has had a real impact on its oral history, North and South America's recent traumatic colonial history notwithstanding.