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The division between Hindu and Muslim, on the other hand, was much more… deeply rooted.

yes and no. I am an indian and feel that i know enought to comment. to put it mildly, india as a country kinda never existed before british (many indian nationalists would hate me for this). if you think of italy post roman empire, all you imagine is lots of small kingdoms, many of which believe in pope, and their is some "italian-ness", but corsica or milan or rome or some other city was independent practically. India was that. before 1000AD, there were hindu kingdoms, but hinduism being poly-theist, there are multiple gods, and most kingdoms had their prefered gods, and they would fight for usual reasons (resources, resources, reesources, and sometimes ego), and even would destroy other's temples, despite both being same "relegion". then delhi sultanate happened, and then mughals, and arguably the closed to todays geographical boundaries of subcontinent closely resembled that of peak mughal empire, but even then they could never get to south-western or southern parts of india, or even the eastern section. was their tension between muslims and hindus - yes, but it was for religious reasons (duh, what do i even want to say) - hindus worship idols, muslims forbid that, muslims are mono-theist, hindus are not. but there were hindus as chief leaders in muslim ruler's courts, and hindu rulers would most likely hate there neighbouring hindu kingdom more than a far away muslim one.

It was britishers who mostly exploited hindus and muslims, and there is always the fringe 10% religious extremist in both groups, who wanted power for themselves, and for that their best bet was supporting partition. fun fact - they both even collaborated in some elections - who in their ideals hated the other's guts. it is always politics. the then biggest party - INC, never supported partition, as recent as 1945, which had both muslim and hindus. it was the acts of extremists who caused riots, which resulted in hatred, but even then, it was not a universal thing. mostly restricted to small regions, roughly close to modern borders. I am purposefully not name calling leaders from either side, one to support my case by hiding how extremists acted, but mostly because it does not help. british did not want to ever even make 2 countries, they wanted 3 plus some independent kingdoms, that way, each country would be small, and they wanted indian countries to be english dominion (much like french overseas territories today - where they would be under the crown, but more independent on internal stuff). And extremists almost immediately agreed. it was inc who fought that and argued for 1 independent country. in the end it just could not happen, because extremists won.

TL;DR - it is not that deeply rooted anger, not 1000 year old, or 500 year old - 100-150 year old, and mostly because of british and extremist. rest 95% would never even care as long as other group agreed to not eat certain meats.

Hindus and Muslims coexisted in the sub-continent well enough before the Brits arrived. There was always tension, same as anywhere else that has millions of people of different ethnicities living together, but the Brits applied the typical divide and conquer technique of all colonial powers and spent 2 centuries prying at those religious divides. There's a reason that the most famous building in India was built by a Muslim.