Most people will not want to move to Mars, just as very few Europeans wanted to move to the New World in the 1500s.
However while relatively few people want to move to Mars, it is a large absolute number. For example a few years ago when Mars One called for volunteers for a one way trip to Mars, they got 200,000 people volunteering. Even filtering for people whose education and skills would make them useful, they had a large number.
One estimate that I saw said that a viable colony would require about 100,000 people to move over a century. And a person leaving Earth can reasonably spend their life savings to do so. The value proposition is essentially sell your house, use the money to move to Mars. SpaceX is planning to make the trip barely above cost. But don't forget. They will be landing on a planet where basic infrastructure, starting with energy, oxygen and tunnels, are provided by SpaceX. That's going to be billions per year. Of course it will be a long time before that colony is self-sustaining. But when it is, those infrastructure basics will be worth a lot.
As for a second contender, developing a rocket that can compete with SpaceX takes over a decade, and many billions of dollars. And then when you try to travel to Mars, you're going to be landing on a place where the infrastructure is owned by SpaceX. Musk is likely to welcome the competition...and charge top dollar for fuel for a return trip. So you'll have to invest billions more. And once you've done so, you'll be competing against an established competitor who is charging thin margins. And you still won't be in that lucrative infrastructure business.
As for who owns it legally...I believe that the people who live there will claim the land that they settle. And will create their own laws. They would mostly come from the USA where legal tradition resolves all property disputes in favor of squatter's rights. There will be room for all for a long time, but guess who the first and biggest squatters would be?