What is the ruling on unintentional lying accompanied by exaggeration in describing a story? Is there a sin incurred by it? Is it considered a false oath by Allah?

Exaggeration in description without mentioning something that did not occur is not considered a lie, especially if it is unintentional. As for an oath that slips from the tongue unintentionally, it falls under the category of a 'laghw al-yameen' (a vain or unintentional oath), for which there is no expiation according to most scholars.