Regardless of the reason, India exists and is united. It could just as easily not remain united (see the Balkans), so its continued existence means something relevant.
And there is shared culture and history across every group within India.
Yes and sort of.
That India exists as a single state now (minus Pakistan and Bangladesh, which at one time or the other have been part of 'India') is enough to make it includable as a single Civ in the game.
BUT the various groups in India do not share some of the most important aspects of culture: language, religion, and ethnicity. My wife and I have friends from India: one family from southeastern India, one from the Punjab in the northwest. One family is Pentacostial Christian, the other Sikh. One spoke Hindi at home, the other a dialect of Urdu. The only language the two have in common is English, and that, of course, is a Colonial product imposed on the subcontinent in the last 200 years. "Shared culture" is only appropriate as a term between them if carefully defined, because many aspects of culture are not in common at all.
On the other hand, I would argue that the differences between them (and other groups in India) are no more or less than the differences among groups in the USA or even the British Isles (get an Irishman, Welshman, Englishman and Scot talking about 'British Culture' if you want to hear an explosive debate!). So differences in the individual 'cultures' of groups within the Civ are not an insurmountable impediment to inclusion in the game.
I think, in fact, that it comes down to a Time Stamp. Modern India is a single Civ. 15th or 10th century India was not. Modern Britain and USA are single Civs, but 'Britain' before the beginning of the 18th century was purely a geographical term, not political or cultural except to help define the differences, and any 'Britain' before the 10th century would be a Blob including Celts (several different polities), Britons, and Germanics with little in common culturally or politically except antipathy. A case could be made that the USA before the mid 19th century (railroads) could be culturally modeled as regions or individual States almost as accurately as a single Civ - and that does not include the Confederacy, which was itself something of a (minor) Blob Civ full of political and cultural differences.
Finally, if we can include the over-arcing Empires like the Achaemenid Persian and Roman, both multi-ethnic and multi-cultural entities, as single Civs in practically every iteration of the game, and 'lump' all of the USA and Great Britain for all time under one all-encompassing roof, a single India is not impossible - it just, like the other instances, doesn't accurately portray the total political, cultural and ethnic diversity of the 'Civ' over the timespan of the game, or even a large fraction of it.
.
