Like most U.S. citizens, I can't help but think of the American Revolution on this topic, and I think it makes a great case study here. After all, you even specifically state that the colonizing country "does not seem to care for [the colonists'] economic interests" despite sharing culture and values with them. There were three major factors that spurred on the American Revolution: there was a widespread belief among British subjects (including the American colonists) that taxes should only be levied against a consenting population, yet the colonists were denied representation in Parliament, and Parliament moved to tax them anyway.
The widespread belief in the idea of "no taxation without representation" represents an enduring social norm dating to medieval England (like a surprising amount of American political and legal culture) and is enshrined in the Magna Carta, a fact that many educated colonists were well aware of. It contains the passage "No scutage or aid shall be imposed in our realm unless by the common counsel of our realm," scutage being initially a tax paid by knights to the crown in lieu of military service but which widened in scope over time. This passage in the Magna Carta established the norm that new taxes should originate in the House of Commons ("common counsel") rather than be levied directly and autonomously by the crown, with the idea that the House of Commons represents the masses and thus provides their consent. King John levied a rather heavy scutage during a period of peace, and anger over this was a contributing factor in the rebellion that lead to the Magna Carta. So, the frustration of the colonists was in some ways a chapter of a larger story, in which English/British subjects repeatedly become frustrated with their government for imposing what they see as arbitrary and onerous taxes without their consent, leading to unrest and sometimes revolt.
In America, much unrest in particular was stirred up by the Stamp Act of 1765. This levied a direct tax against the American colonists to fund the stationing of British troops in the area following the French and Indian War. Many colonists were enraged that this was decided by Parliament over their objections, and argued that it should not have been done while they lacked Parliamentary representation. Supporters of the Stamp Act suggested that the colonists had "virtual representation," i.e. that even if they did not elect their own representatives to Parliament, yet the members of Parliament had interests in common with them and thus could represent them indirectly, just as with most British subjects (roughly 90% of whom were not given the right to vote either).
In response to this position, Daniel Dulany wrote a popular pamphlet Considerations on the Propriety of Imposing Taxes in the British Colonies. I think reading this might make for good inspiration in your case. He argued that even if there were many British subjects that lacked the right to elect their representatives in Parliament, there was nothing absolutely preventing those who lived in Britain from acquiring that right by securing landholdings in Britain, such that even in a population in Britain with few electors, still there would be a few electors in their ranks, and perhaps even a member of Parliament or two. As such, he felt that Parliament would be less inclined to move against those populations, because any such movements would hurt even the electors and members of Parliament among them. The American colonies had no electors or voting members of Parliament at all, so Dulany felt there was no bulwark stopping Parliament from taking advantage of the colonists, because in his eyes the members of Parliament did not truly have interests in common with them, only with other residents of Britain.
Of course, the story of the American Revolution has many chapters aside from this, but just in this brief sketch of part of it we can see a general outline of how unrest can build in a colony not only despite but even because of shared values with the colonizing country. All it takes is a perceived double standard, ideally one that flouts the written law of the colonizing country and widely-held values of its people. In the specific case of the American Revolution, it was a belief that taxes should only be levied against a British population that elects representatives to Parliament, a belief shared by many people back in Britain and enshrined in one of its central constitutional documents. However, it could honestly be anything—anything that leads the colonists to feel that they are being treated as second-class citizens, not afforded the rights they see as their due as subjects of the government of the colonizing country.
As a side note, since you're interested in exploring the case of rebellious colonies that share a culture with the colonizing state, one source of much drama in the tale of the American Revolution is how many of the colonists felt torn between their allegiance to Britain and their allegiance to America. At the time of the Revolution, most of the population of the American colonies saw themselves as British, even quite proudly, and this inspired a great reluctance to revolt in many even if they felt that they had legitimate grievances. Dulany, for example, remained a Loyalist even through the war and had his property confiscated as a result, because he was a passionate British subject and felt that the proper path for the colonies was in protest as opposed to revolt and that reason would win out in Parliament eventually. Ben Franklin was of a similar mindset until relatively late in the game—he was stridently British and refused to side with the revolutionaries until he was dressed down in Parliament in his capacity as deputy postmaster over issues stemming from wide and rather openly rebellious opposition to the Stamp Act by that time (a whole other chapter in the tale). Others like Samuel Adams were quick to the side of revolution and saw less romance in their British status. When the colony shares much in its culture with the colonizing country, you can see that it comes down to the individual how they will react to the prospect of revolution—many will feel a degree of attachment to the colonizing country and everyone will resolve this in a different way. A great source for dramatic character-driven writing!