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Functional Magic exists, is controlled by rhymed verse, and co-exists with normal physics. Matthew, having a good six hundred years' extra knowledge to draw upon in both fields, is enormously powerful by the standards of the day.
God and Satan exist, bringing with them Black and White Morality and the necessity of picking a side. Both forces offer power—evil in the form of Deals with Devils, good in the form of saints—and one must be sure to stay on the good side of your moral compass in order to avoid defeat both in the afterlife and here.
Saint Moncaire, patron of Merovence, brought Matthew here to restore the Balance Of Good And Evil. All the other nations of Europe—Ibile, Allustria, Latruria, etc—have fallen under the reign of evil men, and a usurper, Astaulf, now threatens the throne of Merovence, aided by his Evil Chancellor Malingo. Matthew's job is to find Princess Alisande and help her reclaim her throne, thus preventing all of Europe from falling to the clutches of evil.
A Day in the Limelight: After Matthew disappears from our world, his friend Saul Bremener tries to figure out where he went. He travels to Merovence just in time to be the star of the third book, and becomes the series' only first-person narrator.
Alternate History: The timeline split when Romulus and Remus fought, in this version Remus won and Reme became famous for it's peacemaking and negotiation prowess.
The Atoner: Father Brunel became a priest to find forgiveness for his curse and resist temptation. (He is cursed to transform into a werewolf if he starts feeling lust)
Black Knight: A non-villainous example in Sir Guy. He paints his armor black to hide the fact that he's lordless. Well, as lordless as the heir apparent to the series' analogue to the Holy Roman Empire can get away with.
Mix-and-Match Critters: Matthew meets a "dracogriff" during the second book. The necessary Hot Skitty-on-Wailord Action is deconstructed. Short version: griffons have an estrus cycle that clouds their judgement, and a dragon took advantage. Since both species are otherwise sentient, this is considered rape by everyone who hears about it.
The Promise: in the second book, The Oathbound Wizard, Matthew rashly swears to conquer the neighboring kingdom of Ibile, as his common birth is preventing him from marrying Alisande. The universe holds you to your promises.
Reality Warper: any wizard would count, but special mention goes to Frisson, a genius-savant who comes up with brilliant poetry as easily as breathing... which, given the setting's Functional Magic, can go Off the Railsreal fast.
Rescue Romance: Matthew breaks himself and Alisande out of Malingo's prison and from there it's obvious they will end up together... But not without some entertaining BST along the way.
Similarly, the rightful male-line heir to the ancient Empire of the Latini-Etruscan Federation of old Latruria, and through it Reme, abdicates any right he has to the throne in favor of becoming modern Latruria's first university professor.
Rousseau Was Right: Temptation is all over the place, Matthew actually came close to ending up in Hell, but redemption is also readily available.
Royals Who Actually Do Something: all over the place. Whether villainous or virtuous, there's one thing to be said for the royals of this alternate Europe: they work for their crowns.
True Neutral: Invoked Trope. Saul is a major ally for the good guys and a hero in his own right, but he refuses to accept their religion or belief system in favor of systematically studying how magic really works in this setting, and makes sure to commit a "technical sin" (like eating meat on Friday) for every good deed he does.
Similarly, but for different reasons, King Boncorro in The Secular Wizard does evil things for the good of his country so he'll be richer so he can stop throwing more and more depraved Bread and Circuses at his nobles so he can be rich so he can keep from going to Heaven while not falling into the grasp of Hell- a good example of Stupid Neutral with a heart of gold if ever there was one. His genuinely evil chancellor warping his plans was behind a lot of that.
Succession Crisis: not only is this how some evil rulers take charge, but there's a genuine one at the end of the second book. Of the two people competing for their grandfather's throne, one is the eldest son's daughter, the other the younger son's son. (Solved when the lady marries a third party, removing herself from the line of succession.)
Succubus: Sayeesa from the first novel is a variation, in the form of a Lust Witch. It takes Alisande to break her hold over Matthew.
True Succubus do appear but the book doesn't do a good job at explaining what they actually are; demon, illusion or if they're just other people like Sayeesa.
The Unfrozen Caveman Lawyer: Matthew and Saul. (Later, Matt brings his parents over as well. They are just as proficient at magic as he is.) Having access to centuries of advances and examples in the art of poetry gives them a major advantage.
Unreliable Expositor: A few books in it's revealed that the initial idea of this world operating entirely on Christian theology with Satan and the Saints powering all magic is a local misconception, and in fact All Myths Are True. Although the version of Catholicism practiced in the Europe equivalent is the "correct" religion, Islam is considered close enough and magic works fine even for cultures with no real religious equivalent to the Christian god or devil.