I'm also in for having at least the option to play with a political border if someone wants it. At least I'd play with that.
The main reason is that eventually as is similar here on Earth, any political entity with claims on land won't tolerate trespassing of foreign people, especially military, without an open-border-treaty.
In reality countries respect each others' political borders on a term called "good will". They can't go forth and build a coal mine or set up a new city in another country's territory without treaties or licenses. If a country doesn't respect the borders of another country it always automatically means war; or at least after several warnings have been purposedly ignored.
So the only two situations where military forces are able to wander "freely" through another country's territory is when they have an open-border-treaty, like for example is the case within a military/trade alliance, or when they are at war with each other where they obviously don't have to respect any set territorial claims and mostly are fighting because they don't agree on said border or other issues.
The same is with resources... There might be companies trying to get a license to mine a particular resource in another country, but they can't do so without a license or treaty. They just can't go there with their mining-drills and say "Here I am. Now enforce your territorial rights if you've got the guts!".
I don't see why far off in the future when humanity might be able to colonize other planets/solarsystems and we made the first contacts with other extraterrestrial civilizations such a system where political entities respect their respective political borders on terms of "good will" will cease to exist.
They wouldn't allow us to colonize any planets in their systems without asking in advance and neither would we allow them to colonize any planets within our territory without them asking for permission. They wouldn't allow us to build starbases in their systems and we wouldn't allow theirs. We wouldn't want them to do gasmining on Jupiter without them telling us in advance, and if it's just for the sake of retaining it for ourselfes.
In the end those claims are driven by the need for expansion since existing resources are limited and nobody wants to come too short.
We, as in the entire mankind, may not be able to defend any interstellar claims currently as we haven't progressed far enough in technology to do so, but if we could then we would go to war over any territorial claims like Mars or even other solar systems. If it is for the resources, more space to for people to live in, or just for the fact we think we are the superior race is up to debate and I'm not going that far, but looking back at history mankind has quite a lot of difficulties remaining peaceful because we declared war for many materialistic and stupid reasons.
This particular reasons are why there should be political borders all sides should have to respect when they don't also want to face the consequences involved by ignoring those borders.
As Christian-Akarcro said, the AI (but also the player) should stay out of each others' political territory by default.
One shouldn't have to tell the AI "yeah I don't like that, get out." every single time it trespasses for whatever reasons. The AI should stay out of my political territory by common sense, as much as I stay outside theirs to avoid the diplomacy penalty. Which is generally speaking a joke anyways because the penalty nearly has no impact but in return makes your life much harder because you can't tell the AI that you don't like what it is doing other than declaring war, which I might not even want.
So if it does not stay outside your political territory with its military/colony/constructor ships then this may be considered as an act of aggression/war and the AI should have to declare war on me to be able to do that as much I should have to declare war on it for being able to colonize/invade their planets.
Either that or it may ask for an open-border-treaty to move its fleets through your territory for exploration/expansion/war reasons on the other side of your territory which it couldn't reach otherwise or just to take a shortcut. Maybe there could exist "build"-treaties as well which allows the AI or you to build and/or colonize within each others' political territory if you/they don't mind about it. But at least they/you would ask for that permission in a peaceful way in advance and not just in a "Yeah, I'm here now, so screw you"-way which one can only respond to with declaring war.
So after stating my opinion on why it should be at least an option I'm advancing to how it should work:
Even if it could be directly the Zone of Influence Border I think it shouldn't be that way because that would render the entire game mechanic a bit too easy and non-ambitious.
So either the Political border could be something as easy as a static +5 tiles-border for the planets (Numbers can always be tweaked of course in balancing updates or left to modding), which would mean that the political territory won't be covering everything (which might have it's own appealing gameplay effect)...
There could be a dynamic political border system too based on the influential border, something like a fixed offset (f.e. half) from your influential border which is considered your actual political territory and grows the older your empire gets to undermine "historic claims". (Colors and border styles would have to be adjusted to reflect that of course)
The dynamic political border system could be tied in with the culture flipping... If your political border reaches the opponents' planet and it is still hasn't flipped "peacefully" to that point then the planets' people will stage a bloody coup with the help of some of your empires' underground hardliners and it will flip by force, which of course might result in the opponent getting very angry with you.
At least this would give you and of course also the AI the chance to react on impending cultureflips but with a fixed deadline. If you/the AI can't manage to push the influence back within the given time then eventually a "politicalflip" will happen and end the dispute over that planet. Of course with the political border being half the cultural border it would mean it would take at least twice as long to reach a politicalflip as it would take for a regular cultureflip, but with the gurantee that the planet WILL flip no matter how much influence crap is built on the planet and no matter if the planet is immune tu cultureflipping, so in that sense cultureflip immunity would only delay the inevitable.
I would favor the later, because it would resolve quite some ridiculous problems currently in the game, especially with near-to-nonexistent culturalflips and cultureflipping immunity being way to overpovered, rendering the culture win impossible without relying on military force, which is basically the opposite of what someone trying to do it peacefully wants.
Starbases can't be built within an opponents political border, and neither can he do it within yours. The same should be true when colonizing planets. Outside it's a free for all, even if someone has an influential dominance there, but since nobody has the political dominance nobody has any claims.
Due to war it might happen that two planets close together are being owned by different factions then the political border will run straight in the middle, wherever that is. The influential border might differ quite a lot from that.
If changes to the political border happen due to war/flips, then everything (except other planets) contained within that political territory will change its owner as well, which basically covers shipyards but also starbases, since they can't be moved (which would be actually be better if one could if you notice you are going to lose that starbase when you lose the planet and if they'd work similar to shipyards in terms of needing a sponsor to work to reduce starbase spamming).
I could go on and also suggest something like Endless Space's outpost system, where newly colonized planets are not yet able to have a influential/political borders around them as they are not yet stabilized enough societies/governments to develop a influential/political border around them, which means that they are much easier to flip/invade, and have different diplomatic consequences if flipped/invaded. But yeah, as somebody already said, Endless Space also has "Cold War" diplomatic states between faction, which GC3 doesn't have.
TLDR; I'm for Political Borders as an option one can chose to play with.