Honestly, I feel like the question is comparing apples to oranges. To begin with, "India" in it's current form, as are all African nations, relatively recent entities; states created carved from colonial empires. In India's case there was already an existing empire (empire in the literal sense of the word, a multinational state) which was conquered by many different empires over thousands of years. As whole, India's are used to living in a multinational state and so cultural differences are something they are accustomed to (religious differences, oddly enough, happen to be the biggest conflict causing the Indian/Pakistani split).
African nations are a lot different in that regard. There is little written history in sub-Saharan Africa compared to India, the middle east and North Africa (which has written history for over 5000 years). This can probably be taken to mean that they didn't have large empires on the scale of the other mentioned area that necessitated record keep for tax purposes, foreign debt, etc. That, of course, is pure conjecture. But if we assume that that is true than the fact the the borders were arbitrarily drawn by Europeans (read, randomly drawn by people who didn't live there), that once or twice (or most of the time) they get the borders wrong and the lines go through cities, villages, or even houses.
In other words, the issues in Africa are mainly cause by badly drawn borders. India isn't really in the same situation and the 5000 years of learning to live with other cultures under one empire or another gives them a head start in tolerance.
Since the phrase world's largest democracy is used trendily with India (idk if it is true)
This will be technically true if you believe that either India population is greater than China or you don't believe China is a democracy. Either way, very few people will doubt the veracity of this statement. As for the actual question:
is it the presence of Robust democratic institutions, and what Fukuyama calls precedent against authoritarianism in india the reason for this ?
India is recently a democracy if we look into the history of the region on a long timescale. I have no idea what this "precedent against authoritarianism" is considering that both India and almost all African nations have live under some form of monarchy, be it a tribe leader or a king, for 99% of their history so are not different in this regard. In recent times, India is indisputably a democracy. Most African nations follow the same pattern of government (elected leader, legislatures, independent courts, etc.) as India and are also democracies. I cannot find that this is variable, no matter what angle it is looked at. Furthermore, the idea of democracy and republicanism is not new or invented in the last 200 years. The Greeks and Romans both practiced one or the other in classical times (obviously before the Roman Empire) and they still had no problem with discriminatory practices. In fact, the idea that slavery being a punishment for crime is not even a US thing, it was invented over 2000 years ago where slavery was a common practice in Greece and Rome (and by Greece and Rome, I mean all of southern Europe, most of central Europe, all of North Africa, all of the near east, and much of the middle east and those were all called "Greece" or "Rome" depending on the time period). Granted, conquest in those times were a little different from recent history. After conquering a region, Romans would send out "settlers" so that the new province overtime would become more "Roman" and eventually flush out the local populace. Prisoners (war prisoners or otherwise) were taken as slaves to aid the development of conquered regions (and to perform various tasks as part of the legion, which had 1/5 of personnel as servi).
As a more recent example, the United States most certainly did not try to integrate with the natives in their recently drawn borders for frontier territories. There were attempts at ethnic cleansing early in its history and when slaves were free officially on a national level, many of them were still subject to hate crimes or simply killed in lynching's. There isn't a lot of evidence that suggests that democratic institutions has anything to do with cultural intolerance on the level of genocide. And for completeness, there are obviously examples of non-democracies that participate on genocidal activities, e.g. Jewish genocide (holocaust), Armenian genocide, Uyghur genocide (which counts if you don't consider China a democracy)...