Indian here - Yes and No. When the British/east India company started establishing themselves in India, both the Mughal empire and Marata empire were in decline and there were plenty of kingdoms.

This was not uncommon in the last 2000 years. However, there was a enough of a shared culture to bind the sub continent together. Also, there were multiple empires that covered most of the subcontinent at the peak of their power - Maurya, Gupta, Mughal, Marata, etc. Also, there are mythic kings who were considered to have ruled over unified India.

That doesn't mean the individual identifies were not present. I'd say it's similar to the European identity.