Britain renamed Bharat to India

This is not entirely correct because India as a consolidated nation state did not exist when Brits entered the scene. It was rather a patchwork of empires. This was also the case until right before independence when Congress had to do a mad dash to convince the various princely kingdoms become a part of India by either the use of bribes or force.

This name change is a ploy to appease their Hindu nationalist constitutents and imbue in them a sense of pride and dignity without providing any material benefits. It also alienates the non-Hindu Indians because that's just what BJP does. They have been changing Urdu names to Hindi ones (example Allahabad→Prayagraj) for this very reason. Honestly, I feel if people knew what BJP did (and is doing) in Uttar Pradesh (a province) with Babri Masjid and making it a site for Hindu religious tourism they would understand how venomous BJP's rhetoric and agenda are but it's a whole can of worms.