Neither, of course, is India. It has a great deal more diversity (religious, cultural, linguistic) than the EU, which is in the process of falling apart.
About the only thing India (or Pakistan or Bangladesh) ever had in common was propinquity and common subjection to the British.
Wrong. That's because the only part of Indian history you know of had to do with British India. I can bet you know nothing about the Mauryan empire, Gupta empire, Delhi sultanate, Mughal empire.... each of them lasted much longer then British empire. A north Indian would have a much easier time talking to a Pakistani (an even a Bangladeshi) then say a south Indian (say Tamil). North Indians and Pakistani share same language and cultural habits. Same with Bengalis from India and Bangladeshis.