While you can get a passport to many countries without ever going there, and some may allow you to vote without going there, this is not the big issue.

The big issue is TAX!

Most countries on Earth will tax you if you are resident in them for some part of the year. Some (like US) tax based on citizenship alone.

Martian Independence is not required to avoid Earth tax. Moving there and not being resident in any Earth nation is.

Indeed even in the earliest days of Mars Settlement this should be possible and indeed a good reason to move to Mars - freedom from Earth's tyrannical governments. Its a big reason people moved to America in its early years.

Move to Mars and cease being resident in any Earth country (and renounce citizenship of any country that taxes by citizenship) and you won't have to pay Earth taxes. You can still earn income on Earth from investments that don't tax at source.

It may be worth retaining a Earth citizenship of a country that doesn't tax income produced outside the country.