First, space is huge. Really huge. Generally if you want to put something in orbit and something else is already there, you just need to pick a slightly higher or lower orbit, or occupy the same orbit but some distance ahead or behind.
The most valuable orbital real estate are geostationary orbits, because you need to occupy a fixed point above the equator to avoid moving in the sky relative to the ground. This is a ring of 130,000km of useful space. Maybe it is allocated in blocks of 10km or 1km to avoid near misses, allow for orbital approaches, etc.
Once we split it into blocks, potentially multiple people will bid to occupy an orbital point because it is the ideal point for broadcasting over the US, or over the EU, or whatever purpose they have.
Slots in geostationary orbit could be owned by the first occupying organisation, or claimed by the country on the equator underneath, or shared between all countries on that line of longitude. Slots over the ocean might be owned by the nearest country, or shared between the countries on both sides.
But if there is lots of competition for a slot, someone can just build a suitably large space station to mount all the antennas or sensors required. You might have to rent both the space you need in the station and the frequency range you need to avoid interference with someone else. And your space station can be arbitrarily large, especially in the north-south axis
Eventually you could end up with a 130,000km continuous ring - though this would prevent anyone from having geosynchronous satellites that occupy a geostationary slot or two, but pass through it north and south at the same time each day instead of staying above the equator.
And you could have one or more space elevators connecting to the large stations in geostationary orbit, which also have stations at the counterweight point above them.
The same logic would also apply to a space station at a Lagrange point, except ownership would either be claimed by the organisation that built it, a planet wide governing body, or the last organisation to seize it by force.
For lower circular orbits, it is either a free for all, or you really need a planet wide body to administer claims. 801
It would be very similar to a land registry today. Your claim could reference an altitude, and a phase parameter that would determine where you are in relation to everything else sharing the same orbit. This could be a position at a fixed point in time, and all future positions are then extrapolated from it. Satellites can have different inclinations and you ensure that they cross the equator (and each others orbits) at different times, or more likely everything forms a train in the same inclination, for a much greater density.
The registry body will need to be closely tied to the local military - unregistered objects would risk being destroyed, or the flip side is that if there is no local military, there is nobody enforcing the claims.
At some point it stops being useful to have lots of satellites, and we upgrade to bigger satellites instead. So we have a few rival GPS constellations with small numbers of satellites, but having lots of rival GPS systems just wastes radio bandwidth. We'll have thousands of LEO internet satellites, but there are diminishing returns once any user has a choice of a few providers and can see a few satellites overhead. Space based manufacturing or resorts - the bigger the better. Space based power can also just scale up if the number of satellites gets too high.
Any elliptical orbits would risk intersecting anything in a circular orbit, so a satellite in an elliptical orbit probably has to take responsibility for avoiding anything in a circular shell as it passes through.
Beyond geostationary orbits, there is going to be lots of empty space to go around. You might have some traffic control, but your main concern is probably planetary defence against kinetic bombardment, or smuggling, rather than traffic collision avoidance.
The same logic would apply to any other planet, although they might not spin fast enough for geostationary orbits to useful.
And asteroids are likely just claimed on a first come basis, though larger bodies might be claimed by governments, or could be self-governing. 801