You are right. Most North Indians share more similarities in all those aspects with Pakistanis, than they do with South Indians. Many South Indians share similarities culturally and linguistically and ethnically with Sri Lankans than they do with, say, a Bihari or a Rajasthani. Rajasthanis will find themselves more "at home" in Pakistani Sindh, than in Tamil Nadu. A Bengali would be a lot more comfortable in Bangladesh than in Andhra Pradesh.
But the glory of the Republic of India is that despite all these differences, we can live together as one nation state, and consider all of us as Indians. "Anekta main ekta, yeh Hind ki Visheshta". (Unity in diversity, that is India's speciality.) Hundreds of languages, hundreds of ethnicities and tribes, thousands of distinct cultures, every possible variety of cuisines, "Phir bhi dil hai Hindustani!"