When we start to live on mars, a "Planet of birth" section will be added in our passports

I hope I get to meet the first human born on another planet. That sounds cool.

By the time we have people born on mars we will have had people born in space at all sorts of other places as its doubtful mars will be the first igven the gravity well and lack of resources.

At some point there will be a "first worlds war"

Not necessarily. The current "Place of birth" field already covers that need. If no planet is indicated, like right now, you'd assume Earth, and on Martian passports they'd indicate Mars.