I'm also playing Vijayanana now and I'm well underway to unite all Hindus under my rule. I'm thinking about changing the name of my nation at a certain point to reflect the change of local power to ruling empire of India. Something like Hindustan?
As a hindu nation Hindusthan wouldn't be a very likely name. It's persian for India and is what the Mughals called themselves emperors of. Therefore it's just as wrong as "India" if you're playing an Indian, dravidian country, like Vijayanagar. Bharātavarṣa is a much more indigenous phrase (and it's also much older) for a mythical greater India and a variant of this _was_ chosen for the real life unification of India (the Republic of Bharat is the official name of India).
As other's have said a pan-indian unification in this era might not be that plausible. Also you already have a very cool name as you play Vijayanagar which means something like the empire of the "The city of Victory".
Unification of the "dravidians" under Vijayanagar was almost achieved in real life and might be somewhat more probable as they did encourage the growth of Telugu culture, literature, etc and Telugu was spread over much of southern India as the language of administration. Then again I'm not sure how different that is from just having it as a primary culture. The language's domain did roll back pretty quickly once Vijayanagar fractured.
I'd use the "culture_group_union" command rather than a tag change if I where you so as to not tie it to a certain in game tag if you go the way of a dravidian union.
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