Other thread was closed as I pressed reply, so here it goes.
The reason is that when this anthem was written by Rabindranath Tagore, the communal harmony of subcontinent hadn't been totally shattered by British and RSS nazis yet. It was written when Quaid-e-Azam was still member of Congress, advocated Muslim-Hindu unity to throw out the British, and to include an undivided Punjab, Sindh and Bengal in the Union of India post independence. It was years before he had joined Muslim league, let alone before the word 'Pakistan' came into existence, and was forced to demand separate electorates/autonomy to safeguard political rights of Indian muslims.
Thereafter, things just went downhill. Muslim league kept having deadlock after deadlock with Congress, and despite still pushing for cabinet mission plan up til the mid 40's, the other side (particularly Sardar Patel and Nehru) just wouldn't budge because they actually wanted as many Muslims to leave India as possible so the remaining ones could be subjugated and controlled more easily. Their plan worked, bloodshed ensued and scars linger not fully healed.
So at the time it was written and first sung, it seemed natural to everyone. However, the question you perhaps hint at is whether it was correct of India to keep and officially adopt the anthem in 50s & thereafter. their reasoning probably runs along the lines of massive migration of Punjabis and Sindhis, whose descendants probably number in tens of millions today. Personally I don't see it as any sinister plan to degrade or carry out some design on us in future. We're both separate and happy the way we are, apart from a handful of old, nostalgic grandparents on both sides.
In my opinion, Allama Iqbal's Tarana-e-Hind was a more beautiful song and suitable to be adopted by them. But to each their own, so long as India's anthem is unique and they are proud of it, then I'm more than happy.