India currently defines itself as 'India that is Bharat, is a Union of States'. In other words, India sees itself as 'Bharatvarsha'/'Jambudvipa'/'Arya Pradesh' all of which terms are used interchangeably and which have the meaning 'places where it is permissible to settle without loss of caste'. It was not necessary for there to be political union within these territories- cultural union and religious homogeneity was enough. Similarly, the 'Holy Roman Empire of the German Nation was a collection of States. Even when Germany was unified, the King of Prussia was not described as the Emperor of Germany but as the German Emperor.
Under Muslim rule, the term 'al Hindi' or 'Hindostan' was used to distinguish Muslims born in India who soon developed their own identity. As late as the eighteenth century, a Muslim immigrant- like the famous Reza Khan who held high office in Bengal- refused to accept grants of land because for him Hindostan was 'dar ul harb'.
In other words, there was a scruple re. settling permanently in a less Islamicized country. However, Reza Khan's progeny had no such scruple and described themselves as Hindustani.They would happily move to any place in 'Bharatvarsha' without having the feeling that they were leaving their 'watan'.
In other words, both Hindu and Muslims had the same conception of a culturally and religiously unified India, though- no doubt- at the margin these notions might diverge somewhat.
The British too adopted Indian ideas re. the unity of the country though there were certain deviations from what had previously obtained.
Ceylon was culturally like South India but for historical reasons was administered separately. However, this chimed with an earlier notion that Ceylon was specially dedicated to the preservation of Buddhism. By contrast, Burma was never previously considered a part of India and Indian did not oppose its desire to separate itself in 1935. It is noteworthy, however, that a Hindu Kingdom which could have chosen to join Burma, preferred to remain in the Indian Union.
The question arises- when did India, in more or less its present form, become a unified cultural and religious topos? This is tricky to answer. Going by Hindu religious texts, it would seem that in the first millennium BC, some orthodox people in the West still were suspicious of settling in the East or South. However, other less orthodox people had already settled there and, with the rise of the Samkhya/Yoga school of Philosophy and the great Shramanic religions- Buddhism and Jainism- the arrow of cultural and religious influence was from East to West. By the Third century BC, the possibility of a politically unified sub-continent was turning into a reality because of a pre-existing cultural and religious Union. The Maurya dynasty, from their base in Bihar, are the most notable example. However, Indian legends and mythology already mentioned 'Universal Emperors'who had existed thousands of years previously. The situation in China was analogous. There were legendary Emperors who, it was believed, ruled over the whole of China. This inspired the creation of an actual Empire which would fit that description.
So far, I have spoken only of the 'Sanskritic' conception of India expressed in Indo-Iranian languages. It is now believed that the pre-Aryan civilization of India was very extensive and thus there may have been a unified conception of India as early as the third millennium BC. In particular, the notion that Dravidian languages were related to those of the Indus Civilization is linked to a pre-Aryan cultural unity of the sub-continent. It was thought that Dravidian influence, in particular with respect to Bhakti (devotional) religion was the Dravidian contribution and that this substratum is what has persisted as the elite sacrificial religion of the Aryans fell into abeyance.