The author brings many examples of ancient institutionalized systems of polygamy like in the Inca:
In the ancient empire of the Incas, sex was a heavily regulated industry. The sun-king Atahualpa kept fifteen hundred women in each of many “houses of virgins” throughout his kingdom. They were selected for their beauty and were rarely chosen after the age of eight—to ensure their virginity. But they did not all remain virgins for long: They were the emperor’s concubines. Beneath him, each rank of society afforded a harem of a particular legal size. Great lords had harems of more than seven hundred women.
He also mentions that even in Christendom polygamy was practiced. I don't recall the exact passages, be he talks about feudal lords depleting many of the surrounding villages of young females for "housework" in the castle. He also mentioned a few castles that had secret passageways from the lord's bedroom to the women's quarters, and even the infirmary...
Is this true? Was the supposedly prude and Catholic medieval period maybe a little more mischievous than we think it was?
This isn't true, sorry.
The Red Queen: Sex and the Evolution of Human Nature frankly does not seem like a good source. The author, Matt Ridley, makes extremely broad claims and leaps in logic; he refers to scholars directly and summarizes their findings, but often gives no hint as to the details or context of their arguments or even the title of whatever paper/book he's referring to. For instance:
Laura Betzig of the University of Michigan noticed that, in feudal times, lords favored their sons, but peasants were more likely to leave possessions to daughters. While their feudal superiors killed or neglected daughters or banished them to convents, peasants left them more possessions: Sexism was more a feature of elites than of the unchronicled masses.
Lords killing their daughters? What? And convents were places that surplus daughters could be put, but they also offered great opportunities for freedom and agency.
Moving forward to the specific passage under discussion:
Indeed, in more stratified societies the poor often favor their daughters over their sons. But this is not because of certainty of paternity but because poor daughters are more likely to breed than poor sons. A feudal vassal's son had a good chance of remaining childless, while his sister was carted off to the local castle to be the fecund concubine of the resident lord. Sure enough, there is some evidence that in the fifteenth and sixteenth centuries in Bedfordshire, peasants left more to their daughters than to their sons.
This last sentence has a citation to, again, Laura Betzig, presumably the same place as the previous detail, and it's likely fine. But what about what comes before it?
A large part of it likely came from Ridley's understanding of "feudal" society, which is essentially the uninformed popular opinion of the Middle Ages as a brutish time when lords were movie-villain oppressors. The trope of the lord or king seizing whatever peasant girl he wants, telling his servants "have her bathed and brought to my chamber," and so on is very well-worn - so well-worn that it's the basis of the first episode of the brilliant musical sitcom Galavant, where it is skewered and warped gloriously. The idea of the droit du seigneur or jus primae noctis - the right of a lord to rape a peasant woman before her wedding night - has a high position in the popular notion of the period, but it never actually existed. (Here is an answer on the topic by /u/sunagainstgold and here is one by /u/idjet.) Then the jump from "parents tended to leave daughters property in their wills more than sons" to "peasant women were likely to have illegitimate children while peasant men could easily be childless" comes in, and I have no idea where that came from. It seems like a predetermined conclusion to fit his thesis.
However, there is a seed of truth here in that yes, "feudal lords" were not always sexually continent. There was certainly no hint of formalized polygamy or of harems, but male aristocrats did have mistresses and extramarital liaisons. Betzig actually has an entire paper on this - "Medieval Monogamy" in the Journal of Family History Vol. 20 (1995) - but it is more recent than that citation I mentioned above; still, it may have been closely related to whichever publication the citation refers to. In the very abstract to "Medieval Monogamy", she points out that "in the Middle Ages, as in other ages, powerful men married monogamously, but mated polygynously." The major tension on this point was that the Church was in general quite opposed to extramarital sex. In the early Middle Ages, it was occasionally even a great point of devotion for a married couple to remain chaste! But the Church had enough on its hands just trying to keep aristocratic and gentry men faithful to their wives and eschewing sex workers, dalliances, and long-term mistresses (who were in fact quite likely to be aristocratic and gentry women themselves, sometimes the illegitimate offspring of other men of the same class, as is discussed at length in The Gentleman's Mistress: Illegitimate Relationships and Children, 1450-1640).
You might find this answer by /u/Kelpie-Cat regarding polygamy in early Christian Ireland interesting.