Kashmir was never a part of India, there was no such thing as India before 1947. It was taken over by the Sikh Empire and sold over to a foreigner Hindu Governor and has been under oppression since. They have distinct languages, ethnic cluster, heritage, history and etc... from India.
It's also not true that Kurds have been fighting "for their homeland for centuries". During World War One, they considered themselves an integral part of the Ottoman Empire and gave much blood trying to defend it against the Russian offensive around the Caucasus front. The concept of popular ethno-nationalism in the Middle East is pretty new.
India is free to "expose Pakistan's double standards on international fora", if you wish to piss off most of the world who are opposing the move. Kashmir cause on the other hand, has most of the Muslim world supporting it.