It's unclaimed space. Anyone can plop down a new colony on any ball of rock of they want to. If the colony is established by a government, that planet is now effectively part of that state. The only thing that keeps other people from colonizing that planet as well are the people already on the planet and the people who have funded the colony.
If people want to start a fight over a planet, they absolutely could. But I see little point in it. I established that calculating new routes to new system is very expensive so that players will have to restrict their journeys to the lines on the map. But compared to the overall costs of establishing a new colony, that's a pretty small factor. And it completely pales compared to the costs of armed conflict. Fighting over dirt is just not economical.
If people do fight over planets, and as said earlier, that's assumed to be a very rare thing, then it's about infrastructure and possibly populations. The dirt is the same as in thousands of other systems. It's what people have build on that dirt that's valuable.
If the world population were spread out evenly, we had 53 people per square kilometer. Not completely desolate, but still lover than three quarters of countries that currently exist.
I think I'll probably be changing the numbers, though. More to 90% of people living on the homeworlds and 99% living in the sectors of the homeworlds. That would still be 1 billion people in frontier sectors and provide average frontier sectors with a 5, 4, 3, and 2 million people colony, plus dozens of outposts with hundreds of thousands of people each.