I dunno. The notions of male superiority of our history probably apply to some extent in Halkeginia, but I'm reluctant to think they are the same for the simple reason that magic is a great equalizer. When women are just as capable of murder and mayhem as men in general, some things are bound to change over the milennia of cultural development.
And I'm thinking that bastard sons are unlikely to receive training in magic while legitimate daughters may be walking weapons of mass destruction. If situations such as these present themselves often enough, eventually people are going to stop complaining altogether about female inheritance.
We know Halkeginia still has male priority, but I'm inclined to think that it's generally kinder on female inheritance.
It is, and it isn't.
Magic in Halkegina would be one of three primary societal pressures that would be coming into play to create the society as it exists in that realm. Only two of these pressures existed in our own timeline, it is true, so magic would have some kind of mitigating effect on things, but not as much as people might think because of how European history tended to develop.
First off, the single most important pressure upon society, both in Halkegina, and in our own world, would be the requirement to keep everybody fed and clothed. Because most of the lower classes have only minimal magic--if any--this is something that would only be possible using muscle-powered labor. Halkegina, as it is portrayed, is just beginning to undergo the very first stages of what would later come to be called the Industrial Revolution (or possibly the Second Industrial Revolution, depending on which history you consult), as evidenced by the somewhat widespread use of muskets (as opposed to arquebuses), cannons, and other gunpowder-based weaponry, and the size of Tristania's capital indicates that the accompanying revolutions in agriculture and medicine have most likely started, but the vast majority of the production, labor, and so forth and so on would still be dominated by muscle power, which would tend to mean that males will automatically enjoy an extremely sizable advantage over
most females. Again, this held true in our world, as well as in Halkegina, and would continue to extend even into the ranks of the nobility, since the expectations of the commoners generally dictate the actions of the nobility (albeit in extremely unexpected ways).
The second most important pressure upon Halkeginan society, particularly in the upper crust, is the presence of children, and the relative difficulty in bringing them up to adulthood. This is a major issue for two reasons: first of all because the female population is always the primary bottleneck on any effort for any society to maintain or even expand its population; and secondly, because children, as any parent knows, take up an extra-ordinary amount of time, and if the parents cannot afford a nanny, one parent will likely have to stay home to tend to them. Since, as mentioned above, most labor is still done by sheer physical strength, in most cases, the task of child-rearing will be done by females, since it is something that relies more upon intelligence, empathy, and other more cerebral abilities, rather than upon one's ability to pick the child up, and toss him or her up onto the roof. In short, without television, and several other key technologies, society will typically be either female-dominant, or male-dominant. The simple facts of pregnancy will tend to skew this towards the male-dominant, since pregnancy means several months of decreased female productivity, even with magic, meaning that most societies will still tend to be male-dominated. Basically, what it boils down to, when you get right down to it, is that most societies in Earth's recorded history have been male-dominated for the very simple reason that males are the only ones who have the time to spare to become the dominant sex.
The third major pressure is, indeed, magic. Since both sexes have magic, the typical solution I would foresee is that any partner in a marriage involving magic-users would boil down to whomever is the more magically powerful partner would be dominant within the marriage. Thus, I can easily see Karin being perceived as dominant within her marriage--hence the reason why the servants all agree that it is Karin's fault that Louise's father has turned to Good, despite the fact that he clearly was a
hero, and not a villain, long before he met Karin. But because Karin is the more powerful mage, she is assumed to be dominant. The same will likely be true for Eleanore. Louise will have a somewhat different circumstance--because she is magically powerful, she
should be dominant, but that will only hold true if her magic can be used reliably, which, right now, may or may not be the case. This will have even more interesting ramifications if she actually acts on her feelings towards Henrietta, since Louise appears to be the more magically gifted (and powerful) mage between the two of them, but Henrietta most definitely has the edge in terms of actual political and legal power.
At the end of the day, I would expect Halkeginan society to be effectively male-dominated, but with a strong element of respecting female rights, as well, due to the fact that so many upper-class females can melt your face off if you cross them. Therefore, I expect that the accompanying chauvinism that sprang up around male-dominant societies on Earth would be mostly or even totally absent in Halkeginian society.
I would expect that the primary difference between Halkegina and Earth would lie in how the nobility treats bastard offspring, or even outright commoners who manage to claw their way up from the lower classes. On Earth, it was something of a stigma to be a bastard, and a bastard had absolutely no right to have any expectations of inheritance or any other kind of support from their father...or from their mother, more often than not, since generally the margin of survival for a single mother in pre-computer society was too small for her to be able to offer much, if anything, in the way of material support or inheritance (this was not always the case, however--the first black millionaire in America was a single mother, for instance). However, with magic entering into the equation, things change. The more magic, and the more magic-users, a family has access to, the more militarily/magically powerful that family becomes--hence the reason why the Royal Family would hide some of its magical abilities away, and never teach them to anybody outside the direct line of succession. This would mean that, for the magic-users, bastard offspring are potentially a powerful and very important source of auxiliary power. If I were to hazard a guess, I'd say that part of the reason the Von Zerbst family has survived for so long isn't because they're tough, or devout, or even powerful, but more than anything because they breed like rabbits, and aren't shy about acknowledging bastard offspring. In short, Blitzhart isn't an exception--he's more likely the rule.
Commoners, too, would benefit more from the societal pressures created by the presence of magic, and for much the same reason. The simple fact that magic would be so critically important would mean that commoners would tend to be rapidly absorbed into noble families whenever they expressed any degree of magic, and a family of commoners that consistently produced magical offspring would likely find itself elevated to the status of nobility whether the rest of the family wanted to be nobles or not. For those commoners who resisted being absorbed into the nobility, the simple fact of their magical (or even non-magical) power would act as a cloak of respectability that no noble family would wish to look down upon, lest it come back to haunt them. This is why one of the original four Council members had to be popular with the common folk: if the commoners didn't support the Regency Council, the Council itself could not hope to draw upon the magical or the material reserves necessary to prevent the more powerful nobles from going off and doing their own thing. With de la Mott gone, the Council will find itself being forced to resort to increasingly harsh and unpopular means to maintain control--means which will inevitably mean that Louise will gain increasing amounts of support as she moves more and more openly within her home kingdom.
And before you start to point out Louise's attitude towards commoners, I would simply point out that Louise's attitude towards commoners is actually pretty mild, compared to the attitudes shared by much of Medieval and Rennaissance Europe. Louise may not have much respect for them, but at no point is she seriously rude to them (without provocation, anyway)...and her opponents
are.