I'm thinking of this more as a concept than as a discrete time in history. Since, in general, Western nobility would attempt to marry partners of similar status, it's logical that over time the number of connections between people increases. Even today people keep track of how closely related spouses are among the remaining royalty and gentry. Has anyone written about such a concept before and how it might have influenced interactions in a courtly setting, or the environment of other noble gatherings?
This is a really interesting question. At least in the Middle Ages, my area of concentration, there is quite a bit of recent historiography on this topic. In sum: the older view might be keen to examine the noble family in purely diplomatic terms, according to alliances, politics, and feuds. This narrative left little space for female agency, or (as you say) a willingness to examine the nobility as an actual family.
Recently, some scholars, most notably Amy Livingstone (see “Out of Love for my Kin: Aristocratic Family Life in the Lands of the Loire, 1000-1200”) have taken a more nuanced approach to aristocratic families. She emphasizes the aspects of noble family relations that we in the 21st century might recognize - arguing that parents might have actually loved their children, and that females may have had more agency than previously thought, among other things. I personally like Livingstone’s work, because, as I think you were getting at in your question, we shouldn’t view the nobility as machinery. These were people, and though it may be tempting to assume that their cultural and political ideals were representative of the way they actually lived, things were probably quite a bit more complicated than this. Though at least for the Middle Ages, the sources we have make it difficult to answer this question. At least people are trying!
EDIT: BTW, I don't think Livingstone has a chapter specifically on the court, but she would be a good place to start!
The short answer? Yes, it very much was like that, especially in Europe where countries were packed close together. Though for female royalty that were married to form alliances and sent to a different country, it was generally accepted that they may never see their immediate family again.
If you were, for example, Marie Antoinette, you could not leave France to visit your home country of Austria because it would become a diplomatic nightmare. While to us it may look like a sister visiting her brother, this brother happened to be the ruler of Austria. The French people would have viewed it as a potential betrayal for their adopted Queen to leave the country even for a visit, and Austria may have viewed it as a potential invasion. The only time Marie Antoinette saw any of her immediate family after being sent to France was when two of her siblings came in an unofficial capacity and in disguise.
Political relations between countries and families were extremely tenuous and could and did change constantly depending on the turn of politics. If you were European royalty, almost every country in Europe would have been headed by one of your relations. Since everyone was related, politics at the time were intensely personal and were swayed greatly by the personality of monarchs. You can see how, though having friendly relatives in distant courts could be a positive to your political power, it also became difficult when your country went to war with a country your relatives were running. Many foreign royals were constantly divided and torn between their loyalties to their home and adopted countries, and those run by their relations.
If you have any questions about specific periods or places in history, I'd be more than happy to enlighten you or point you toward further reading. My era of knowledge is anywhere from 1500-1900, and especially the great power houses, dynasties, and royal families of Europe.