It would appear that pointy hats were quite common in many parts of the world including Japan. Although the one we usually think of is the 'dunce's' hat given to schoolchildren to wear.
In the witches' hats article the writer mentions 'Cone of Power' which I thought a tad grandiloquent and ultimately meaningless. But perhaps not. Conical hills are landmarks that stand out, often with the aid of a tower. If Michael aka Hermes is associated with high cone-shaped places, tall hats could be a pagan throw-back.
Dunce, meaning 'a stupid person', is said to have referred to followers of the Scottish philosopher John Duns Scotus who was famous for splitting hairs and rejecting new Renaissance-style ideas. But why pick on Duns and not Scotus? Surely the English would have delighted in a Scotus pun denigrating their Scottish neighbours.
The approved headgear for dons is a completely level mortar board, the opposite to a cone shape.
I don't know about Japanese dress code, in such a status-conscious society perhaps the higher the hat the more important the wearer? Japanese guides in London who wish to stand out generally brandish a certain insurance company's umbrella.