For India, Kashmir became its integral part when Instrument of Accession (Jammu and Kashmir) was executed by Maharaja Hari Singh on 26 October 1947. According to which Jammu & Kashmir will be a part of India (with some terms & conditions), so naturally it should be India's part.
For Pakistan, Kashmir has become a Muslim dominated state, so naturally it should be their part as the whole partition of India & Pakistan was based on Religion (Hindu & Muslim).