This is somewhat outside of my area, and we're looking at an empire that officially lasted about 300 years – although real Mughal power lasted to the mid 18th rather than the mid 19th century. So I'll try and identify certain mechanisms that set Mughal social organisation apart, especially under Akbar (1542-1605); and in a second part turn to some more concrete cultural examples from the mid 17th century. Regarding your premise, it seems difficult to compare a certain region's diversity with that of another – although an interesting aspect of the Mughals is certainly their openness to South Asian faiths, as well as to Persian culture and (to an extent) to European influences. This can be connected on the one hand to how various other Islamic realms allowed their subjects to follow their religions, albeit in conjunction with a special tax: similar examples (but by no means using the same tactics) include Islamic Spain with its dhimmi rule and the Ottomans. On the other hand, my professor used to highlight more generally how throughout South Asian history foreign rulers, including the Mughals would invade the subcontinent, but then over time adapt themselves to local customs and organisational forms.

Social/ religious organisation: The Mughal emperors were Turco-Mongols of the Timuris dynasty, and were all Muslims. While Babur's conquest founded the empire, Akbar is usually described as consolidating Mughal rule in India (I suppose hence his byname “the Great”). He emphasised the Great Mughals' dynastic charisma, his own spiritual leadership as well as his law-giving capacities. For administration the Mughals relied on an organised military elite who formed part of a hierarchy of service. Most of such civil servants were Hindus, although Persians would also increasingly play a role as administrators and especially as scholars. Akbar also reformed the revenue system, compiling a fixed average for price and revenue data. Furthermore a system of hierarchical classification (mansab) of all military and civil officers was adopted, relating offices to salaries and promotions to higher offices to fiefs. Hermann Kulke has argued that this system was highly advantegous under Akbar, but would contribute to the Mughal's eventual downfalls due to being automated and too rigid.

Turning to religion and law-giving, Akbar followed the ideal of the just ruler that had precedents in both Muslim tradition (with the mahdi) and in the Hindu concept of dharma (connected to legendary king Rama). However, Akbar wanted to add legitimation not only through religion, but also by giving laws. This idea in turn was less acceptable to his more orthodox Muslim subjects than to his Hindu subjects. Muslims disagreed with Akbar's new belief Din-i-Illahi (Belief in God) and his reserving final decision in religious matters to himself. To quote Kulke: “[Akbar] wanted to establish a synthesis of all religious ideas that appealed to him and prevent sectarian strife a a supreme umpire. He openly opposed the orthodox Islamic scholars (ulama) whom he castigated for their medieval outlook. Akbar's bold attempt at creating a new, tolerant religion died with him; but the idea of a divine grace which was bestowed on the Mughal dynasty and constituted its charisma remained alive” (p. 190). Parallels to his inclusive attitude can be drawn with the Sufi's mysticism. Despite the failure of his Din-i-Illahi belief, a focus on diversity connected to the civil system and its Hindu civil servants would prevail under following Mughals.

As you mention dynasty, it should also be added that there was no clear line of Mughal succession which strangely enough had a stabilising effect: Victors of power struggles (often between various sons of one emperor) would often try to reconcile the vanquished leaders' supporters as to gain legitimacy. Such was the case with emperor Jahangir who rebelled against his father Akbar. Apart from organisational parallels, one interesting addtion under Jahangir was the stronger influence of Persian culture, due in part to his Persian wife Nur Jahan and her entourage. A further blending of Persian and Indian culture took place under Jahangir's son Shah Jahan, under whom the empire would experience its 'Golden Age', evident in the architectural innovation (the most famous being the Taj Mahal).

Intellectual exchange: Education also played an important role in upholding Islamic traditions while paying strong attention to Indian traditions: Under Akbar mosque schools (maktab) and colleges were built, the latter teaching Islamic theology, jurisprudence, philosophy, logic and astronomy. Then again, there were also schools and centres of higher learning for Hindus that looked to ancient knowledge, including Hindu theology, Sanskrit grammar, philosophy, literature, logic, astronomy and mathematics. What is more, at Akbar's court a cultural renaissance took place, with the production of important artistic and literary works such as historiography and Hindi poetry. These efforts included a department of translation with works in Sanskrit, Arabic and Turki translated into Persian, which was understandable to both Hindus and Muslims. Even a translation of the Quran into Persian was undertaken, although I'm not sure how unique this was at the time – as the holy word in Islam was and is Arabic.

I'll now turn to a later period, the mid 17th century during the last years of the emperor Shah Jahan's rule (see a map of the empire at the time here) – to give different examples, but also to trace intellectual developments begun under Akbar to later generations. The 17th c. was a period of important intellectual exchange in India, to quote Ganeria: “Muslim, Jaina, and Hindu intellectuals produced works of tremendous vitality, and ideas circulated around South Asia, through the Persianate and Arabic worlds, and out to Europe and back”. This atmosphere found expression in 1656 in a project by the Mughal emperor Shaj Jahan's son Dara Shukoh: Drawing on works by Akbar's chronicler Abu al-Fadl he initiated the translation of 52 Upanasids (central texts of Hindu philosophy) into Persian, for which a large team of bilingual translators was assembled. Dara's goal was to show that the differences between Hinduism and Islam were for the most part terminological, and that the Upanasids could even function as a form of commentary on the Quran – ideas in keeping with his forebears' religious policies. Another important school of thought at the time were the Hindi 'new reason' (Navya Nyaya) scholars, who focused on reflection, evidence and proof in a way that for Ganderi recalls contemporary European empiricists, while drawing on Sanskrit traditions at the same time. 'New reason' training centres flourished and attracted visitors from all over South Asia and beyond.

The influences going to Europe and back are also evident at the Mughal court at the time. One example of this is the French philosopher Francios Bernier, who came to India in 1656 and came in contact with Dara Shukoh's translation project. Bernier mentions that he translated rationalist works among others by Descartes into Sanskrit only a relatively short while after their European publication. Among other Europeans active in India were the Italian Roberto Nobili and the German Jesuit Heinrich Roth who studied Sanskrit as well – both wrote works on Indian philosophy and grammar that were not published until recently, thus hindering a more thorough transmission of ideas. I sketched some developments in this part to show the possibilities created and actively encouraged by the Mughals for transcultural exchanges between various Islamic, Hinduistic and even (more limited) European/Christian traditions, on religious, philosophical and literary matters.

Summing up: The Mughals were successful for quite some time at managing an empire that incorporated various ethnic and religious groups, for reasons both unique to their context as well as connected to older traditions. First, their religious policies should be seen as drawing on Islamic precedents that made relative religious freedom of the subjects (while paying special taxes) possible. Second, policies such as Akbar's reliance on Hindu civil servants have to do with Indian “Realpolitik”: As a Muslim empire with a large Hindu population and with neighbouring Hindu realms whose powers varied over time, the Mughals had to in some way accommodate Hindu influences. This drawing on local elites was standard procedure in many other parts of the world, e.g. in forms of European expansion. Jahangir's employment of Persian officials on the other hand led to stronger Persian influence in the realm, in connection with Persian being a sort of lingua franca of Islamic powers at the time (maybe similar to French at contemporary European courts). In addition to these social changes connected to exterior influences, the Mughals again starting with Akbar also took up an expensive program of cultural patronage, accompanying religious diversity with a mixing of influences from Islamic and Hindi arts. The results from these programs had a legitimising function as well in that they communicated the successful “syncretism” taking place that was connected to the Mughal rulers' religious “charisma” to the outer world.