The interesting thing about India is it does actually refer to more than just a sovereign country as of today. Its equivalent is Europe where the Europe consists of different races but are united in the their unique European culture and history where they have borrowed from each other extensively as well as good dollops of foreign influence in terms of knowledge and culture for example by the Arabs and Turks.
What is ironic is that probably present day Pakistan is more deserving to be called India than say other parts like the South or NE which are no where near the river Indus. The word is corruption the British used originating from the river Indus and the term used then Hindiya/Al-Hind by the Arab explorers. The Persians referred to anyone living around that region as Hindus/Hindis. So did the Arabs. Now please note that Hindus was a religion neutral term, it was strictly associated with geographical terms of people living east of the river Sindhu/Indus. Similarly the Arabs referred these people as Hindis regardless of whether they followed Islam, Buddhism e.t.c
It was only under the British rule that Hindus became associated with a religious term. I remember some lecture in which Zakir Naik himself that as long as the term Hindu is taken as the geographical context, then "I am a Hindu".
Of course the communal affects of the British rule and the partition later on have created such an affect that Hindu has been reduced to a religious term. I guess people in Balochistan or other areas of Pakistan that are far away from the Indus valley could be ambiguous, but the Punjab and Sindh that cradle the Indus river would ironically be definitely have been referred to as Hindu/Hindi regardless of WHO ruled over them. Just as for example when the Arabs ruled of the Persians, they automatically didn't become Arabs.
Here is an interesting read on this:
The Meaning and Origin of the word Hindu