The geographical ideas of the middle ages which were just ending at the start of age of exploration included the idea of the Three Indias.
The three Indias included the Greater India which started at the Malabar Coast (in modern India) and points east. Thus it included much of modern India, Bangladesh, Burma, Indochina, Indonesia. and maybe Australia, the Phillipines, China, Japan, etc.
The Lesser India ended at the Malabar Coast and thus included much of modern India, Pakistan, and maybe alot more of southwestern Asia.
The third India was apparently father west, and may have included East Africa.
So the Portuguese explorers considered every place they visited east of the Cape of Good Hope to be in one or another of the Three Indias.
Similarly when Columbus sailed west and discovered islands, he assumed that he was somewhere near the islands of east Asia, and so in India the Greater.
And the title that the kings of Spain used as rulers of their colonies in the Americas was "King of the East and West Indias, the Islands and Mainland in the Ocean Sea".
So naturally all people of non European ancestry descended from the pre 1492 population of that vast kingdom of the Indias were called Indios meaning Indians. It was not because of the initial mistake of Columbus, but because for centuries the offical title of the Kingdom which claimed all of the Americas said it ws the the kingdom of the Indias.
The Spanish may have considered to the continents to be geographically the Americas, but they considered everything west of the border with the Portuguese claims to politically part of India.
And apprently in English "Indias" became "Indies" for some reason. Even though the East Indies are part of the concept of India the Greater, and even though the West Indies were all part of the KIngdom of the Indias at one time or another.