The problem with saying “X language is not indigenous to Y place” is that it just depends how far back you go. France, Spain, Portugal and Romania had Latin enforced upon them, and what about Patois and Nigerian Pidgin where they’ve developed to the point of being different languages, spoken only in the place they developed, or Singlish where nobody forced English due to the circumstances surrounding Singapore’s colonisation, but it was a common language spoken by all of the various ethnic groups there and picked up features from some of the other languages spoken there?

If Indian people don’t refer to themselves as Indian when speaking English, then that’s just a characteristic of Indian dialects of English.