"Once upon a time" is one of the openings one finds for folktales, particularly when published and "toned down/abridged" for children in English editions.

The folktale was the novel of pre-modern folk. Generally intended for adults, they could be extremely elaborate - taking several nights to tell, and as works of fiction, they were set in a fabulous time and place, the fiction of which, was emphasized by initial phrases like "once upon a time." It is akin to "a long time ago in a galaxy far, far away."

Besides folktales, the other well-known folk genre of narrative was the legend, stories generally told to be believed. Because the legend emphasized reality, it was often linked to real places and times. "Once upon a time" would be the exact opposite of what the legend demands. For the legend, the beginning would be more like "a few years ago in the village next to the village in the south, ..."

Although you do not address it, the folktale often concluded with a standard ending - best known to English readers as "and they lived happily ever after." As with the introduction, this could be different given the language and place; in Irish folktales, a common ending was “Tá sé go máith, agus níl sé go dona”: “It is good, and it isn’t bad.”