Classical mythology of the ancient world can be regarding as having two genres of legends (stories generally told to be believed) that have counterparts internationally. The first of these are aetiological (or etiological) legends, which explain the origins of things. These are fantastic in nature and usually do not correspond to scientific explanations of how things originated, but because of their nature, they are often primordial or at least prehistoric in the time that they describe, so we can set these aside as stories that do not have a counterpart in history.

What we're really talking about are those stories in mythology that can be regarded as historical legends - that is, stories about more recent times and about people that were presumed to have lived in the past. These were oral narratives, and as such, they were inclined to drift away from any real events and people that may have inspired them. At the same time, many historical legends internationally are likely to be based on nothing in any real past, while others can be surprisingly accurate descriptions of past events. It is, consequently, a mixed bag, and each example of mythology must be approached by historians differently.

As a body, ancient mythologies can be used as valuable primary sources, reflecting the cultures that recorded them, and to a certain extent, casting light on previous times when the stories circulated orally. That does not make them "history" but it does make them valuable to historians.

At the same time, there are those who have attempted to use ancient mythology as a means to peer into an unknowable past - into past events not accessible with the written record and perhaps only hinted at with archaeology. These efforts are often scorned because they attempt to take the stories in mythology too literally, stretching the limits of credibility. In some ways, your father is absolutely correct that "the present form of 'History' as we know [it] is mostly a 'western' way of recording their pasts, which is mostly incompatible with the history." - at least history as it has been conceived in modern professional studies.

There is also the simple fact that some cultures invest a great deal of effort in maintaining stories exactly the way they were told, while others delight in changing stories with each telling. The former are more likely to pass on information with some accuracy about past events than the latter. I discuss this problem when approaching oral tradition in my recent book, The Folklore of Cornwall: The Oral Tradition of a Celtic Nation (2018).

You also may be interested in this recent discussion dealing with your question from another angle, which includes myself as well as /u/Anubisbreath3000.