If you trace the history the root of the problem was originally muslim and hindu rivalry, it's often called "communalism" in subcontinental politics.
The Indians that were advocating for an independent India expected to inherit a unified India stretching from Pakistan to the East to to Bangladesh in the West. However, the muslim minority in India were afraid that they would be made into second-class citizens which is why they advocated an independent muslim Pakistan under Jinnah. The name was dreamt up by Mohammed Iqbal, a muslim poet and a politician. These fears were largely groundless on the whole - despite the violence of partition which mostly was the fear of what could happen. It also turned out that the misgivings of many observers who felt such a large-scale experiment in democracy could not work were misplaced - it did work - India took yo democracy lile it was born for it - it encompassed all sorts of shades of opinion from communist Kerala and West Bengal to Brahminical New Delhi. However, one might say with the rise of the RSS and the BJP in recent years, their fears were well-founded although almost eighty years later. Perhaps the Indian politicians who banned the RSS should have never lifted it. But then again, India never experienced what fascism was like in Europe, so perhaps it felt it had less to fear, or did not know what it should fear. Time will tell.
On independence the status of Kashmir was not resolved and so muslim-hindu rivalry became the Pakistan-India rivalry around the disputed status of this territory as well as the rivalry over their respective cricket teams.