No, and it is a good thing too. This is prevented by the Outer Space Treaty. If it weren’t for that treaty we would have nuclear weapons in orbit, for sure. We’d also have pointless military bases on the Moon - the Apollo mission would instead have been a military mission to establish a base there, just to show that they could, and Russia would have joined them soon with their own base there somewhere, to lay claim to the Moon. They might even have had shooting matches between each other on the Moon. Then by now China would have joined them for sure (it could have done a lot more to get into space if there was some prestige / military objective)..
Such a present or future would be the end of space exploration. Habitats are extraordinarily vulnerable. Spaceships travel at kilometers per second, far faster than a high velocity rifle bullet. Faster than ground penetrating missiles too. There is also nowhere a human can live for more than seconds without a spacesuit anywhere on the Moon, Mars, or anywhere else in space except the Venus upper atmosphere - where you need protection from concentrated sulfuric acid instead of protection from vacuum. Combine those together and if we had any warfare in space at all, the “rebels” would not hide in caves. They would just all die as soon as anyone hits their habitat with anything, even just a rock.
So, we can’t have territory in space established through military means like we can on Earth. It just wouldn’t work. We need another way forward.
However the Moon itself is pretty much worthless, there is nothing growing there, same for Mars. Is only valuable as far as you make it into a habitat. So the habitats are the things that really matter. You already own your habitat according to the Outer Space Treaty - anything you make yourself.
There are other ideas also - a safety zone around the habitat for instance, such as they already have around the ISS. Also there may be a way to establish “functional rights” within the OST, such as that the owner of a mine can continue to operate it so long as they keep it in good order, but loses rights over the mine if they abandon it.
So there are ways forward.
My own sympathies with the idea of some form of resource sharing, and such ideas are common here on the continent. The idea is not that you distribute everything evenly amongst everyone. It is that you have some recognition that you are mining the resources for the benefit of everyone, of course still making a profit on them and you sell them on return to Earth.
It could take many forms including a requirement to assist other countries to do their own mining in space, transport and equipment, or it could be some kind of a tax - I personally support the idea of a tapered tax starting at zero but going quite high if we get trillion dollar industries in space. It then would be used to fund a sovereign wealth fund for the entire world similar to the Norwegian sovereign wealth fund or the Alaskan one
Or else to provide a world base income that everyone receives similar to the system in Alaska. I am especially concerned about the effects on the global economy if we have our first trillionaires in space, with wealth equal to that not only of a small country but of the US, able to pay off the US national debt out of their own pocket. I think that billionaires are fine but trillionaires could be too powerful and unbalancing for our economy unless something is built in to deal with that situation in advance.
I think that it is very important to get such a framework in place first . To do it later once we have trillionaires in space, if that happens, may be very hard. They may be able to blackmail and buy out entire governments. Or they may keep prices of materials from space artificially high, for instance the diamond cartel that keeps diamond prices much higher than they need to, they could keep prices of platinum from space far higher than is necessary. Just low enough to destroy the Earth's platinum industry, then once that is achieved, raise prices and keep it as high as they can when it could be as abundant and useful as aluminum.
Space trillionaires, if such is possible, could also act against any possibility of humans in space long term. After all their main interest would be financial - at least - the ones who make most money from mining space would be the ones who win out, and that may be best done with robotics. They may siphon all the money from space into building a luxury mansion in Hawaii made of gold and platinum or whatever it is that tickles their fancy. While if this future is planned for right away then if it does happen then it might be, for instance, that we no longer have poverty to the level of people going hungry and unable to pay for simple medicine or the most basic of education anywhere in the world. And it could also be that the sovereign wealth fund, part of it is used for both human and robotic space exploration and indeed maybe for such things as setting up those lunar cave colonies.
I'm not talking about the idealists who have set up the first mining companies, but the later bosses that take over those companies or that set up rival companies and beat them at what they are doing once they can see that it is profitable to do so.
That's why I'd be happy with a 0 level of tax right now, so long as somehow the mechanism is in place to be able to do something about it later. I think now is the time to look forward to the future. There are many ways to do it. We are only able to explore space in peace at present due to the Outer Space Treaty without which we'd have nuclear weapons in orbit, and quite possibly useless token military bases on the Moon and the US, Russia, and now China all spending large sums just to hold a claim on the Moon. Back then they had a lot of foresight setting up that treaty and we need similar foresight now I think as we move to the next stage. As for the detail, well it will surely involve lots of negotiation amongst politicians and spade work by lawyers which may seem boring and unnecessary but I happen to think that it is very important.
You can't establish a "stake" on the Moon. That's not possible within the OST. Yes private individuals could lay a stake, say "I own the entire Moon" even, but their governments as signatories of the OST have agreed not to make any property claims on the Moon so they can't support those stakes in any way. So it is just like you pointing at the Moon and saying you own it. You could do that but it wouldn't have much meaning since you can't assert those rights that you claim.
Some other system would be needed. Possibly based on habitats and a safety zone, or based on functional property rights.
The ideas about either sovereign wealth funds, or a guaranteed world income for everyone, etc - they are completely consistent with the largest investor getting most from the investment. They are ideas that are already used in Norway and Alaska and the companies that do the mining are big oil companies and get the lion's share of the income.