Just like in the title, it amuses me that for such a vast and rich subcontinent UK can rule over all of them when every other powers want it too, shown by existing ports of European powers already in India. When EIC eats and destroys various Raj and Maratha Empire bit by bit, why no one takes the chance and get their own portion of land too?
Non-English European powers had colonies all over India, from the 15th* through the 19th centuries.
Portugal controlled the entire island of Ceylon, as well as having colonies all along the west coast. They held Goa from 1509 until 1961.
The Dutch East India Company controlled the southwest and southeast coasts, and had territory in modern Bangladesh - they also managed to take Ceylon from the Portuguese and hold it until the end of the Napoleonic Wars.
The French held territory on the Coromandel Coast, the Malabar Coast, and in Bengal, which lasted until the 1950s, technically.
Denmark-Norway controlled minor territories in Tamil Nadu, West Bengal, Kerala, and the Nicobar Islands until 1868, when they sold Tranquebar in Tamil Nadu to the British.
Though dated, K.M. Panikkar's books Asia and Western dominance, 1498-1945 and Malabar and the Portuguese: being a history of the relations of the Portuguese with Malabar from 1500 to 1663 are pretty good sources for the history of Western colonization in India.
*Technically. The Spanish were granted land in India by Papal Bull in 1493, but had it rescinded in 1494