This is the familiar story of the birth of the United States (except the last part).
In other words, history will repeat itself.
One explanation why it might unfold in this particular way is that, until recently, the US seemed destined to conquer and dominate the space by itself, elbowing out competition - just like it did with (most of) the North American continent.
Looking at the space rush that has just started, the picture is more complex.
For starters, there are at least two other major actors : China and (to a lesser extent) the European Union, each with its own history and mental models of relationships with territory and population. Second, there will be no Earth government in the foreseeable future.
Consequently, the history of the relationship between space colonies and the originating political-territorial entity will most likely be determined by what the source is. In addition to history and mental models, another factor is the contribution of the private enterprises to space exploration and colonization.
So how the future political configuration of space might look like?
Most likely, on each major celestial body there will be a patchwork of territorial units, with various degrees of political integration with - and representation within - the political system from their respective place of origin, some of them fully independent, others protectorates or provinces of Earth nations; let's not forget penal / correctional colonies.
One indicator of full political integration is whether space-born folk can acquire positions of power on Earth, and vice-versa.
Let's look in turn to each of the three space powers, in order to imagine what can be currently expected from their colonies.
US
The private sector has a strong and potentially dominant contribution, therefore some colonies might be chartered, organized, and run like company towns, while others along principles of direct democracy, but all having in common (at least initially) a low level of involvement from the US government.
Contrary to the image derived from American history, there will most likely be no fight for independence, because all these settlements will be independent from the start. On the contrary, at one point there might be a significant movement to join the United States as outer-space states (Texas?), or (in the case of company-run colonies) to have it brought under the jurisdiction and authority of the US government in order to stamp out abuses and violation of workers' rights.
China
The government is in firm control, and private ventures are in the service of the country, and are allowed to operate to the extent they make China richer and stronger. Institutions, instruments, and techniques of control will be there maybe even before the first people, so integration with the motherland will be strong and not be allowed to be questioned (Hong-Kong).
All territory acquired through Chinese efforts and sacrifices is seen as perpetually part of the indivisible China (one China), so uncontested independence is out of question (Taiwan). If there will be a war of independence, it will be extremely violent and destructive, and the only way the colonists could prevail would be to incur a crippling blow to terrestrial China that would relegate it to a second-rank power.
Europe / EU
Things are more complicated here, primarily because the fragmented colonial history of various European countries. I nevertheless expect a fully integrated space citizenry, with all the democratic rights and responsibilities, and enjoying an economically comfortable life, without extreme inequality, and supported by a strong social security system. How this might shape mobility between European and US, respectively Chinese colonies, remains to be imagined.
Of course, other citizenship models can be retrieved from history, like some of the imperial ones (French / Spanish ?), where all subjects were nominally equal, and could travel, work, or get an education anywhere across the empire, but non-metropolitan natives were politically restricted to their birthplace region.
Before that, Europe should settle the citizenship issue (currently only bestowed by Member States); it might be that the first European citizens are born somewhere on the Moon or Mars.