In "War Zone," a caption identified Mars as an "Independent Member of the Earth Alliance." I'd interpret that as meaning that not every iceball and spacestation run by humans under the auspices of the EA is a "member" of the Earth Alliance, like Proxima III or Orion VII or Earth itself, and those members can encompass more than one planet (so, for instance, the Senator from Proxima III would also represent the Proxima IIIa moonbase, and the space stations in orbit of the planet, and the mining outpost one jump away with a crew of thirty), so up until the end of Season 4, Mars was directly governed by Earth, or, less likely, the Earth Alliance as a whole (similar to how Babylon 5 had an Earth-appointed EA officer as military governor), but post-independence, they had sovereignty over their own planetary government and could hold elections for their leadership (or, I don't know, appoint a king, if they wanted).
Or, to be more topical, Mars had been like Puerto Rico or Washington, D.C., but after Sheridan strong-armed the EA, it became a regular state.
I could see it as more like a Commonwealth of planets. A former Earth colony world that is now independent, similar to UK nations that used to be UK colonies. The head of state would be the president of the EA.