This is a really complicated question that has unfortunately received a bit of politicization which I think is important to clear away a bit before actually answering your question. But before I do I need to stress that Indo-Aryan and Dravidian are both language families and not culture or ethnic designates. There is certainly a degree to which they align with ethnicity, and states founded by Dravidian speaking peoples and Indo-Aryan speaking peoples had distinct, if heavily intertwined, histories, but that does not make these useful ethnic categories.

The reason I specific this is that Indo-Aryan is a branch of the Indo-European language family, which originated in the west Asian steppe and spread outward starting around 6000 years ago. The Indo-Aryan branch, which includes Hindi, Farsi, Bengali and more, reached the subcontinent about 2000 BCE or so. This became politicized because the spread was originally characterized as an invasion of Indo-European steppe nomads which caused he downfall of the Bronze Age Indus Valley Civilization. This view is now discredited, in no small part because of the complete lack of evidence for this invasion, and the current view is more along the lines of a series of migrations. It is possible, although by no means certain, that the subcontinent was largely peopled by Dravidians speakers, and those in the north adopted the Indo-European language. Genetic evidence overwhelmingly argues against a population replacement.

This has been politicized by two fronts: colonial British scholars liked to use the linguistic connection to portray the Indo-Europeans as conquerors who brought civilization to the subcontinent, and viewed it in highly racialized terms. On the other hand, Hindu nationalists have tended to latch on to the overwhelming wrongness of this hypothesis to discredit the entire idea of Indo-European migrations, constructing detailed arguments that either Indo-European originated in India or that Dravidian and Sanskritic languages (eg, Hindi) are actually related, neither proposition has any linguistic support. It makes the matter a rather delicate one.

So what did happen? The earliest states in the subcontinent arose in the northern plain, the arya-varta that encompasses the basins of the Indus River and the Ganges. These states were made up of speakers of Sanskrit, a Ind-Aryan tongue (actually, by this time they were largely speakers of Prakrit, or regional languages descended from Sanskrit--through a somewhat odd turn of events Prakit literature is in many respects older than Sanskrit's, its linguistic parent). Through a very complicated process including empire building and long conversations, the culture of these states spread far outside the basin, including the "Hindu" religion. Independently and a few centuries later, states developed in south India of speakers largely of Tamil, a Dravidian language. The culture of the north was highly influential in the south and these states were "Sanskritized" during the first centuries CE. This was very much a two way road, however, and particularly during the middle of the first millennium distinctly south Indian cultural forms spread north. Even today there are certain cultural distinctions between north and south India, but overwhelmingly they are defined by cultural borrowing, dynamism, and innovative contact between the two.

A good source for this is Romilia Tharpar's History of Early India: From the Origins to Ad 1300.

Dravidian and Indo-Aryan (Indo-Iranian) are language groups, not clans or cultures. There’s some overlap of language with culture, but on the whole, there are huge commonalities of culture across India despite the presence of scores of languages.

India has had extensive and prolonged population mixing over thousands of years, so demarcations between “separate cultures” are fairly weak, especially among the Hindu population, because Hinduism is very old and has had a long time to mix and homogenize.

You can see this in languages. Indo-European and Dravidian are two completely different language families, but if you look at the actual languages contained in these families, there are remarkable similarities. For example, Tamil and Malayalam are Dravidian languages, but literary Tamil and Malayalam have vocabularies that are about 70% derived from Sanskrit. This is strange, considering that Sanskrit is the base of Indo-Aryan languages in India which is a totally different language family. It goes to show how thoroughly people have mixed in India.

Or consider Hinduism. It arose as a synthesis of Vedic and Shramanic traditions in north India, but quickly spread to the south. The thing about Hinduism is that it isn’t monolithic, it’s a mix of different sects. And many of these sects aren’t even north Indian to begin with. Shaivism, for example, probably originated in south India and then spread to the north. Because of the very long history of Hinduism in the subcontinent and because of the plurality of beliefs allowed within Hinduism, all parts of the country including north and south have contributed different traditions, which have intermixed and been subsumed under the umbrella of Hinduism.

Or you can look at it from the perspective of genetics. There has been extensive intermixing of all Indian populations, north and south, for thousands of years. Genetics studies show that for the first two thousand years after Indo-European languages arrived in the subcontinent, people intermarried freely across all regions. Around 1900 years ago the caste system became more rigid, and the intermixing decreased. But by that time Hinduism was already well-established, and the caste system didn’t prohibit marriages across language or regional lines, it only made distinctions based on caste.

It’s always been a mixed bag. There are forces of integration and separation at work simultaneously. Lots of things are common across India, but different parts also have their own unique features, language, literature, cuisine, local beliefs and customs. There are always regionalist movements going on somewhere or the other in India – someone thinks their religion is under attack, someone thinks their culture is being neglected, someone thinks their language is not accorded the respect it deserves. But they also have a heck of a lot of shared culture and experiences uniting them.