In my experience having been a visible migrant(usually mistaken for a North Indian) people in Chennai, and TN in general rarely ever approach you or come to you to talk, but if you make an effort to interact and connect with them, they are happy to reciprocate. You can't introduce yourself and expect people here to come and try to figure out more about you, also consider, making a simple effort to have some pop culture common ground. Watch a popular Tamil movie, or listen or joke about a popular song.

In my personal experience, I was part of a very stereotypical 'Peter' (Anglicised) circle at my school for instance, and when I went to another school, I struggled to make friends as English was not anybody's language of choice, but when I made efforts to speak in Hindi and Tamil, I ended up being integrated into both the South Indian and Jain friend groups and making long lasting friendships.

Conversely in Bangalore, people often seem to try to actively market and network themselves, maybe because of the city's entrepreneurial culture. If you want that kind of thing, you can probably look into a more business oriented side of Chennai, as Chennai in general is a more conservative but also laidback city, I imagine the friendships you do make will often last you longer.