This is a false understanding of the differences between the two countries (actually 3, Bangladesh is also part of this shared culture). It's not 1000s of years of history being
opposed to each other but 1000 years of shared history and culture. The differences have only arisen in the last 100 years or so.
Pakistanis share common languages with only a part of India, the main languages being Hindi/Urdu and Punjabi. India also has roughly the same number of muslims as there are people in Pakistan. People outside of the Hindi belt in India might not speak Hindi/Urdu or Punjabi, regardless of their religious identity. Historically speaking, people of the subcontinent have identified more with their ethno-linguistic identity than with their religious identity, no matter what the politics of these countries as well as the talking points of the politicians of these countries might make you feel. Evidence of this fact can be seen in the reasons for the existence of Bangladesh as a country. They separated from West Pakistan on the basis of their Bengali Identity. Similarly in India today, there is a strong movement in South India against the imposition of the Hindi language.
When it comes to music, there are two broad schools of classical music, Hindustani Classical that is from the North of India and now Pakistan, and Carnatic music that is predominantly from the South of India. Hindustani Classical music has always had a shared contribution from Pakistani and Indian signers. Many Pakistani singers (such as Rahat Fateh Ali Khan, and Atif Aslam) have contributed significantly to this school, as well as to modern Bollywood music, and there are many such artists in India as well (such as Zakir Hussein). Many Indian lyricists write Hindi/Urdu songs for Bollywood as well (such as Gulzar).
Arguably music and arts have always unified the two countries while other factors have caused rifts.