We made the mistake of not conflating East India Company with Britain, and it didn't end well for us.
a) what was Christopher Columbus looking for?
b) what was Vasco da Gama looking for?
c) what were British/Dutch/French East India Companies looking for?
d) what did Alexander's ambassador Megasthenes write about in his book Indica?
e) where is Mahabharata set? Bharat is still one of India's official names.
India has had a shared cultural history for a very long time.
That it wasn't always called India is a moot point.
But being a single governance group isn't the point. Being culturally similar is what defines India. And it obviously is.
What we are discussing is foreign ownership and foreign power over India, and you just said that allowing EIC too much of these was a mistake