You seem to be essentially asking how the borders of post-colonial independent nation-states relate to the borders of colonial states. The answer is that the borders of post-colonial states overwhelmingly follow the borders of the colonial states that they descend from, and that the borders of colonial states were often quite arbitrary and based on geopolitical contrivances (take a look at the Caprivi strip in northeastern Namibia for an egregious example). I don't think you're right in assuming that Britain and France consciously and maliciously divided people of the same ethnicity in some kind of divide et impera grand strategy. This was true in certain local instances, but if we look at colonialism more broadly, I would say that a much bigger problem was the exact opposite: peoples who were vastly different from each other in many ways, and who might have been better off with their own sovereign states, were forcibly incorporated into larger colonial states where they formed one of many groups competing against each other for resources. Nigeria, for example, was created by forcibly merging many different ethnic groups like the Edo, Igbo, Yoruba, Hausa, and Fulani into one vast state with no historical basis whatsoever. Pre-colonial polities in the area we now call "Nigeria," such as the Edo kingdom of Benin in the south and the Sokoto Caliphate in the north, had very little in common and had entirely different languages, social structures, belief systems, etc.

So coming to your specific question "What caused India to be united whilst other areas were not?", the simple answer is because the British had succeeded in establishing their rule over many different groups of people in the Indian subcontinent, and their sovereignty was merely transferred to the newly-independent Indian Republic in 1947, following the Indian Independence Act. Burma became a separate country because it had been separated from the Dominion of India by the Government of India Act of 1935, while Pakistan (including Bangladesh, which was originally East Pakistan) was separated from British India following a successful campaign by Indian Muslim separatists in the 1940s. But you seem to be interested also in the regional identities contained within the Indian subcontinent and their political relevance, and since Indian history is an area in which I have a special interest, I will write a mildly long post on that as well (hopefully, others don't mind).

Prior to the British colonization of the Indian subcontinent, the subcontinent was divided into a number of distinct regions which had their own distinct identities (often, but not always, corresponding to ethno-linguistic identity). Examples of such regions include Gujarat in western India, Bengal in eastern India, and Andhra in southern India. Each of these regions had their own distinct language (Gujarati, Bengali, and Telugu, respectively), their own distinct alphabet or writing system, and often their own distinct castes or social groups as well. Each of these regions were also frequently under the hegemony of a single dynasty, and even when they weren't, they were still conceptualized as a single region. For instance, following the collapse of the Kakatiya dynasty in 1323 (which had united the Telugu-speaking lands, and whose rulers called themselves "Lords of Andhra") there was much chaos and infighting between various Telugu lords, but in spite of this political disunity there was still a conception of an "Andhra country" (Andhradesha) defined by common language and culture. This is clearly seen in records left by local Telugu lords, such as those at Srirangam by Mummadi Nayaka in 1358 and those at Annavarappadu by Kataya Vema Reddy in 1403. Both of these local lords conceptualized themselves as being rulers of the Andhra or Telugu country (despite in fact only ruling a portion of that country), because they saw that territory as a natural unit with a common identity.

The importance of these regions can be gauged by their endurance as discrete units even after conquest by Muslim foreigners, and even following incorporation into a larger empire. For instance, the regions of Gujarat, Bengal, and Andhra were all conquered by Muslims, but even after Muslim conquest, Muslim regimes in these regions largely retained political boundaries that corresponded to ethno-linguistic (and hence, regional) boundaries. It is no coincidence that there existed Indo-Islamic states like the Gujarat Sultanate (1407-1573), the Bengal Sultanate (1352-1576), and the Golkonda Sultanate (1518-1687) whose boundaries largely conformed to those of particular ethno-linguistic regions. Each of these regional sultanates were conquered by the expanding Mughal Empire during the 16th and 17th centuries, but the boundaries of these states were preserved as subah (province) boundaries, as seen in this map of the Mughal Empire.

I should be careful to not overstate my case. Not all kingdoms in India corresponded to distinct regions with their own identity. For instance, the sultanates of the western Deccan (such as those of Ahmadnagar and Bijapur) did not follow any particular logic in their boundaries; they simply came about due to infighting in the erstwhile Bahmani Sultanate, which controlled the entire western Deccan up until the late 15th century. There was no "Ahmadnagari" or "Bijapuri" regional identity as such (in the same way that there was a Gujarati or Bengali identity). However, even when there existed states that failed to correspond to particular ethno-linguistic identities, the people living under those states (or at least, the native elites) often had a sense of their own ethno-linguistic identity, as the Maratha elites serving the Deccan sultanates certainly did, and these native elites had their own mental maps that went beyond the ad hoc boundaries of these sultanates.

When the British began arriving in India in increasing numbers during the 18th and early 19th centuries, and began gradually establishing their rule over various parts of the subcontinent (a process that is too complicated to go into great detail here), the ethno-linguistic identity of Indians continued to remain relevant. The manuscripts collected by the Scottish army officer Colin Mackenzie in 1802, for example, indicate that "India" was perceived as consisting of 56 countries defined primarily by language or ethnicity, and that "[T]he people in these countries, though they profess different religions have different family names and customs yet are known or specified according to the language they speak; as Tamils, Telugus, etc." (T.V. Mahalingam, Mackenzie Manuscripts, Vol. 2, p.320) The British, for their part, did nothing to discourage native languages and literature; on the contrary, many of the British in India took a great interest in Indian languages, and individuals like C.P. Brown made great contributions to Indian literature.

At the same time that literature in regional languages (and hence ethno-linguistic consciousness, at least amomg the literate classes) was rising, there was simultaneously a great homogenization of educated elites across India thanks to the pan-Indian spread of the English language (which had completely replaced Persian as the language of administration and education by the 1830s), and hence an increase in pan-Indian consciousness among these elites. The Indian National Congress, established in 1885, was the first truly pan-Indian political association in history. Over the years, people from all over India - Gujaratis like M. Gandhi, Kashmiris like J. Nehru, Bengalis like S.C. Bose, Tamils like C. Rajagopolachari, and many others - would join its ranks. What united them all was the English language and a shared sense of socio-cultural identity as members of the Anglicized elite of British India. Although all of the people I named above were well-aware of their regional identities, their identity as "Indians" was more politically relevant (whether or not the ordinary masses identified as strongly with the "Indian" identity as the Anglicized elite is a quite separate issue), and with a few exceptions (notably Periyar E. V. Ramasamy and the Dravidian movement), none of the educated English-speaking Indians pushed for ethno-linguistic separatism. The most relevant separatist tendencies came instead from the Muslim separatists of the All-India Muslim League, whose demands for a separate Muslim state in the Dominion of India was satisfied by the British in 1947.

After Indian independence, however, there were several prominent actors who demanded ethno-linguistic states within India, but not secession from the Indian Republic. The most notable of these was probably Potti Sriramulu, who demanded that all Telugus of the erstwhile Madras and Hyderabad states be united into one new state. He fasted to death in 1952 in support for this cause. The next year, the Telugu-speaking Andhra State was created, and in 1956, the States Reorganisation Act drew new territorial boundaries for Indian states based on ethno-linguistic boundaries. By the 1960s, most of the Indian states that exist today had been created, and many of these (like Maharashtra, Gujarat, and Karnataka) have borders that were determined by the geographic distribution of certain ethno-linguistic groups. Thus, New Delhi effectively created "nation-states within a nation-state" as a way of satisfying demands by ethno-linguistic regionalists without compromising the integrity of the Indian union.