In Harry Potter and the Deathly Hallows by J.K. Rowling, is there any mythological story behind each of the Deathly Hallows?

The Deathly Hallows are three magical objects central to the plot of the Harry Potter series, especially the last book. Though the Hallows are real, physical objects in the story, they are surrounded in myth and mystery. Something which is hallow is holy or sacred, and while it's an unusual form of the term, a sacred or holy object could be described as a hallow. To call these three magical items--a wand, a stone, and a cloak--the Deathly Hallows tips us off to the fact that they are related to death and are sacred for that reason.
The story behind the Deathly Hallows goes something like this:
Three brothers were traveling through some woods when they came to a river too deep to cross by wading through. Luckily, the three brothers were wizards and easily conjured up a bridge. Death (personified) was angry that he had been cheated out of three souls, so he appeared to the brothers. Death congratulated the brothers on having outwitted him and offered them each gifts of their own choosing. 
The first brother, who loved to duel, asked for a wand more powerful than any other. Death grabbed a branch from a nearby elder tree and transformed it into a wand.
The second brother, who was arrogant, wanted to humiliate Death and so asked to have the power to bring back the dead. Death grabbed a stone from the riverbank and enchanted it for the brother.
The third brother, the smartest of the bunch, asked for the power to walk free from this encounter with Death unable to follow him. Death gave him his own cloak, which granted the brother invisibility.
The three brothers then crossed the bridge and went on their ways with their gifts. The first brother made it into a town and challenged another man to a duel. Of course, his wand was the most powerful in existence and he won. He then boasted and dared anyone to try and take the wand from him. While he slept that night, someone crept into his room, slit his throat, and took the wand. Death had outsmarted the first brother and given him the key to his own death.
The second brother made it all the way to his own home before trying to use his enchanted stone. He turned it over three times in his hand, and before him appeared the figure of his dead fiancee. Though he could see her, he could not touch or speak to her. The second brother was so distressed that he killed himself, joining his long-gone love, the first brother, and Death.
The third brother enjoyed his gift for many years. The cloak gave him the power to remain invisible to Death, and only when he was ready to go did he take it off and bestow it upon his son. The third brother greeted Death like an old friend and finally went with him.
 
That is the bulk of the story, but we learn from Harry's friend Ron that there is a little more to it. Ron and lots of other wizard children grew up hearing this story as a bedtime story or folk tale. Though it is somewhat ambiguous, we are lead to believe that the Tale of the Three Brothers actually happened when it is revealed that the Hallows are real items. It is implied that the invisibility cloak Harry received anonymously during his first year at Hogwarts was the hereditary gift of the third brother.
With regards to the conflict of the seventh book, it is believed that any one person who owned all three of the Deathly Hallows would become a Master of Death. Owning the wand, the cloak, and the stone would make someone an essentially unstoppable force. As such, Voldemort wants to get his hands on the Deathly Hallows and fulfill his dream of immortality.
https://harrypotter.fandom.com/wiki/Deathly_Hallows