I am currently taking a Medieval Indian history class. Obviously the British East India company is mentioned as it precedes the fall of the Mughal Empire, the last topic we discuss. In my time I have seen the British, Dutch and French East Indian Companies. Why was there so many and how did they all end up with the same name? Seems recent enough for them to have named themeslves or were the named based on what they were doing like the Dutch West India trading company
The different national companies were corporate entities that their respective governments granted monopolies on trade to the East Indies, whether that be the islands or, later, India itself.
They were East Indian to clarify that they operated in the East Indies, and their names were all slightly different - the British Company was technically called the Honourable East India Trading Company, whereas the Dutch one was the United East Indian Company (Vereenigde Oost-Indische Compagnie) and the French one was the French Company for commerce in the Oriental/Eastern Indies (Compagnie française pour le commerce des Indes orientales).
Monopolies were valid within countries, but there wasn't, as far as I am aware, any particular expectation that, say, the British Crown would prevent its own traders trading with India because the French Crown had granted a monopoly.
So there were multiple national trading companies vying with each other, hence the different national names.