Why can't non-white characters just act like regular characters?
Is it absolutely necessary to play up the fact that she is Indian and acknowledge it at every story point?

Unless the show wants to go that route, okay (see Blackish). But it is perfectly fine to have a non white character that could just as easily be played by a white actor.
New Girl is a good example. Winston and Coach don't need to constantly assert and acknowledge their blackness. They just need to be funny characters.