I can only talk about the Indian case, and hope someone can give more insights into the takeover of Congo (which I'd like to learn more about). First off, something I'd like to add regarding your premise of a shared “narrative”: To me it seems that comparisons can be useful when they make sense, i.e. when the cases are in some way similiar enough to merit comparison. In this case we have not only two very different colonial powers and situations, but the takeovers are also 50 years apart. For the rebellion of 1857 I'll discuss its precedents and consequences, as the events of the rebellion themselves seem less relevant to the question.

Precedents: The rebellion took place 100 years after the beginning of the rule of the Easty India Company (EIC) in South Asia, and has to be understood as a continuation of a long tradition of local resistance to colonial rule (that continued up to independence). In this light, the rebellion was particular in its scope, but not in its goals and demands. There were many different kinds of and motivations for resistance: Among others religious insurrections, banditry, acts of revenge – but also peaceful opposition. Most of those insurrections were connected in some way to the very common economic exploitation and political marginalisation through the EIC, and thus to popular discontent. It's also interesting to note that such discontent ran through all social classes. As a common ideology we can identify the central role of the community principle to the rural populace (including the caste system), and that of traditional land rights. Most (if not all) of these motivations for and forms of resistance played major roles during 1857. On the other hand, the lack of a coordinated, unified organisation meant that the British could count on parts of the local population, and was a major influence in the rebellion's failure.

Consequences: While I'll skip the rebellion's major events, it's important to mention that the atrocities committed on both sides, as well as the fact that insurrections had sprung up in most parts of the subcontinent (although mainly in northern and central India) made a return to the status quo unthinkable. The official British reaction was the takeover you mention: It meant the abdication of the last (Muslim) Mughal ruler, and the end of the (Hindu) Maratha realm and of EIC rule in India. What is more, rule was transferred to the British Crown, in this case to queen Victoria, effectively creating “British India”. From now on, India would be ruled from a Foreign Minister from London, supported by a council with 15 members. The former General Governor was made Viceroy, and concessions were made to popular opinion by including more European and Indian members in the councils of Bombay and Madras. More generally, Indian policies started playing a bigger role in British parliamentary debates.
Other major changes included the larger British military presence in India, a better coordination of military and government, and restructuring of administration and army. A “feudalisation” of the former EIC possessions was advanced through the strategical furthering of certain lords and land owners. This would influence a further division of colonial society, due to the increasing separation between colonial hierarchy and Indian population.

As I said, I'm not qualified to compare these changes in detail with the takeover of the Congo. However, I'll try and highlight major differences between the two cases that come to mind:

Causes - As I've tried to show, 1857 built on a long tradition of popular and elite resistance, and the wide-spread and extremely violent uprisings were the main catalyst for the takeover. In contrast, in Congo it were the Belgian monarch's extreme exploitation and mistreatment that led to Belgium taking control.

Time periods - In Congo a period of 9 years lay between exploration (1876) and annexation through Leopold II, with the takeover taking place in 1908. In India the British had a long tradition of trade and military incursions starting in the early 17th c. before first taking control in Bengal in the mid 18th c., and then 100 years later replacing the EIC with crown rule – we're talking a much larger frame of time here. In addition, 1857 was at the starting point of the phase of British “New Imperialism” with an even stronger racial component, whereas during the time of Leopold's Congo in the late 19th this newer wave of imperialism was already in full swing.

I could go on, but what I want to stress are not only the many difference between the two takeovers, but mainly the nature of these differences owing to different points in time and two very different colonial and indigenous traditions. After all, the Congo was Belgium's only colony, in comparison to the huge British empire. Similarities I could pick up at the moment are: With both there was misconduct, by Leopold II and the EIC, as perceived by the governments and (depending on the case in intensity) the population of the metropolis. And with both there were important structural parallels before and after the takeover, especially regarding continuities in administration, but also continuities in exploitation (of resources and people). As mentioned at the start, such similarities seem not conclusive enough to me to construct a shared narrative between colonial takeovers.

Sources: On resistance in colonial India see Ranajit, Guha, "Elementary Aspects of Peasant Insurgency"; and Chatterjee, Partha, "For an Indian History of Peasant Struggle", in: Social Scientist, 16/11, 3-17

On the rebellio of 1857 see Bayly, C. A., "Indian Society"; and Markowitz, C., "A History of Modern India, 1480-1950

Edit: Grammar