If we get wars in space between space colonies, they would be soon over, with no survivors, because the colonies would be tremendously vulnerable. Surrounded by vacuum, there's nowhere to go. No hills, caves, forests where you can hide and survive because there is no air to breath. And rockets by then would have to be commonplace and just a rocket coming in to land but not slowing down would destroy a colony, with velocities of many kilometers per second needed for space travel. Even tens of kilometers per second - and faster than that if we have really fast travel like days instead of months to get to another planet.
So - either we colonize space peacefully and co-operatively, seems to me, or not at all. You could have hardened military based dug beneath the surface of the Moon say, but they'd have to be quite deep also to not be affected by an impact by a multi-ton incoming spaceship traveling at kilometers per second. So - that's without any weapons at all, just peaceful technology turned to war purposes.
Anyway - for now, then the outer space treaty prohibits owning any celestial body. And all the space faring nations have signed it.
Also on the plus side - outer space is pretty much worthless for colonization. I mean the actual land itself, surface of the Moon, other planets, asteroids. It's only of value when made into habitats. So that's very different from Earth. There's no great value in owning large strips of vacuum covered landscape. The habitats would be the assets in space, and many of those I think would be in free space.
And - we just have to be more peaceful than we are now, to have permanent settlements in space, or they will all be destroyed pretty quickly. At least in space we have to be more peaceful.
We've made a good start with the OST, and we are currently exploring space in a spirit of exploration, as pioneers. At this stage there is no real reason for any competition except of the friendly sort, competition in science, seeing who can find out the most, do the most prestigious missions etc. And I think myself the longer we can continue at this stage the better.
Like Antarctica. Though there are many nations there they are collaborating as scientists and the place is of little value for settlement. Space is of even less value for settlement than Antarctica, at present anyway, despite all the hype. Is nowhere in space remotely as habitable as Antarctica or the Earth deserts or the sea floor.
If we find a way to mine the asteroids economically that might change - I mean not that it becomes more habitable, but that there is so much to be earned from space activities that this pays for settlement there - like the mining towns in deserts on the Earth - and I think there might be multiple issues to be sorted out there if that happens. Also mining for ice and water. Who owns it - according to the OST then nobody does, and we mine it for the benefit of all of humanity. But how is that actually going to work out once we have commercial companies returning resources from space to the Earth and to other locations in space? I think that's the first big hurdle we have to face.
I think prospect of anyone trying to take ownership of areas of planets or Moons is likely to be a long way down the road. But in between might have the situation where there are valuable habitats that have been constructed, with disputed ownership for one reason or another. That could lead to conflict perhaps. E.g. a multi-square kilometer Stanford Torus and then some dispute arises amongst those who built it about who owns it or some such. I think potentially could be tricky things.
Also - resources from space - if we start returning so much by way of resources that it destabilizes the Earth economy e.g. that platinum from space is totally controlled by a few small companies. I can imagine that leading to tension between countries, even between companies as wealthy through space activities as the richest nations on the Earth in competition with the Earth itself.
If so we will have to sort that out. Otherwise - well we probably have to stop space exploration. Which is a pretty major incentive to somehow find a solution.