For Germany (German-speaking lands):

How did Europeans during the medieval and early modern eras view Islam and how aware were they of the ties between the Abrahamic faiths?

German Christians in the century before the Reformation, and Catholic Christians during/after the Reformation would have a pretty good chance of learning about a theological connection between Islam and Christianity.

I can't speak to how much attention any individual paid on their own, and I'll discuss an important exception. In general, though, the opportunity was most likely theirs for the taking.

No, nobody's writing a thorough, neutral treatise On the Faith of the Turks or whatnot. Thomas Burman even argues that some of the Renaissance-era Qur'ans from Latin Europe were basically made as luxury display objects for bragging rights. But Muhammad, now--the person of Muhammad is a different story.

Dante's Inferno is many, many things, but one of them is a hit piece on his political enemies as well as traditional villains of Church history. And in the middle of this, he sticks Muhammad into the circle of hell where heretics burn. (Well, in this case, is eternally ripped in half over and over, with his intestines hanging out, but I digress).

Note: "heretics." Dante doesn't supply any details on this point except to point out that the "in half" part is because Muhammad caused a 'schism' with the Church. In other words, a split of something that was once whole. Dante could use Muhammad this way because he knew his readers would recognize the name and the man's significance as the central figure of Islam.

Two wrinkles. First, the Divine Comedy is Italian, it's in Italian, and it wasn't translated into Germany until the late 16th/early 17th century.

However, Italian and German literary culture was surprisingly intertwined in the 15th/early 16th centuries. (Printers in each region are printing texts in the other language, even.) Basically, German cities like Nuremberg and Augsburg desperately wanted to be Florence and its compatriots. Printer-writer Christopher Scheurl, for example, insisted that the brilliant humanist writer (and badass nun) Caritas Pirckheimer was Nuremberg's answer to Venice's star "learned lady," Cassandra Fedele. I'm not saying that Germany's literati were reading the Comedy itself. But they were part of the same general pool of ideas and knowledge.

The second objection, though, is the big one: the "literati"--elite literary culture--was a super duper tiny part of the population. By 1500, we're looking at negligible literacy rates in rural areas (the vast majority of the population), and even in cities maybe 30%--and of course, a lot of those people are just interested in reading stuff for work, maybe a pamphlet or two predicting the end of the world, and religious-themed clickbait.

Basically, a percentage of the population so small it seems barely worth mentioning. But I want to start with it to establish a baseline view of just how much late medieval Christians were Not Fans of any Abrahamic religion besides their own. Muhammad is seen as a heretic, as the chief heretic, as a false prophet, and sometimes even as the Antichrist. He’s the bad guy.

Moving beyond the elite: there are two major ways that your average illiterate peasant could have learned about Muhammad, and Islam’s direct connection to Christianity. Both of them come through the same medium: sermons.

By the fifteenth century, we know now, the Church’s longtime lip service to reaching as many lay people possible had actually become a commitment, and rather a success! For centuries, people had been required to memorize the Creed, the Our Father, and eventually the Hail Mary and Ten Commandments in order to stand for the baptism of their children/godchildren. These would most likely have been learned within the home. And indeed, the Church would continue to lay major responsibility for religious education on parents.

But by the fifteenth century, public sermons came into their own as a vehicle for religious education accessible to all. In Germany, they were even typically held after Mass, often outside the church building itself, and evidently people often came even if they weren’t particularly interested in the Mass ritual itself. They were definitely preached in the vernacular (German) as well, and were mostly lessons based on an interpretation of that day’s standard Gospel reading. (Some advice books for priests even just basically say, “translate the Bible into German for your audience”).

From the late 14th century especially, an increasing number of priests did their preaching with the aid of books—sermon collections. However, medieval sermons as written tended not to be the complete speeches we think of. They would be more like skeletons, maybe even still in Latin. The preacher would have to translate, to cut out parts he didn’t find relevant—and to our point here, add in little stories to illustrate various points, called “exempla.”

One of the major sources for exempla was a very large collection of stories about “saints et cetera,” let’s say, called the Golden Legend. It was hands-down one of the most popular books of the 15th century—both in clerical hands, and in lay people’s libraries. And kind of randomly, mixed in with stories of early Church saints and a history of the Lombards in Italy, there’s a Life of Muhammad.

It’s exactly as inaccurate and polemical and Islamophobic as you think, of course. But in keeping with the idea of Muhammad as a heretic that we saw with Dante, the Golden Legend goes into a surprisingly amount of (wrong) detail on Muhammad’s religious knowledge before he wrote (we’ll get to that “wrote”) the Qur’an and about some of the (warped ideas of) devotional and ritual Islamic practices. Of particular interest for this question, Jacobus actually uses quite a few examples that trace a Judaism-Christianity-Islam relationship:

In his laws he put some things of the Old and New Testament. For when he was in his first age he haunted Egypt and Palestine, and was a merchant and led camels, and conversed oft with Jews and with Christian men…

…And because that the Jews worship towards the west, and the Christian men towards the east, therefore he would that his people should adore towards the south

Do I know for sure that preachers drew an exemplum or two specifically from this chapter in the Legend? Of course not. But it is entertaining, and entertaining in a manner that complements rather than doubles up with “God gets revenge on Jewish people” or “this is how the beautiful Christian virgin was graphically tortured.” So in my opinion, there’s a nonzero chance parts of the exemplum made their way into a sermon or two.

And now we arrive at the Immaculate Conception.

This is the Church doctrine that holds Mary was conceived and born without guilt from original sin. It wasn’t officially adopted by the Catholic Church until the 19th century for reasons. But by the early 15th, it was basically all the rage in medieval Christianity…except for the Dominicans. You know, the religious group literally named the Order of Preachers.

In the 15th century, preaching was a real competition among parish priests, Dominicans, and Franciscans. The Dominicans opposed an Immaculate Conception; the Franciscans were zealous proponents of it. And they wanted to teach it, and by all accounts of popular devotion, the laity were quite eager learners. (Mary was a big enough deal in late medieval Christianity that there’s a whole subset of tactics that preachers used to insist that no, Mary was not actually a goddess).

Especially after the establishment of the printing press as a German institution around 1470, a lot of priests had access to a collection of skeletal sermons that they would customize. The second-most popular one has the most excellent title Dormi secure--a reference to its purpose of letting priests sleep easily at night because they’ve got a cheat sheet for the next day’s sermon. It was even more popular with Catholics in the 16th century, even. And it very much comes from Franciscan hands. Right near the front is a sermon dedicated to arguments in favor of the Immaculate Conception.

And one of them is, because Muhammad said so in the Qur’an.

No, really. Listeners learned that the saraceni agreed with the idea that Mary was conceived without sin, and that “Muhammad [praises] her in his “alcoran.” With primary source quotes to back it up.

True, the actual quotes that Christian preachers cherry-picked and interpreted…generously…are hadith, not verses from the Qur’an. And “alcoran” can take on some creative alternate spellings in some manuscripts. But one way or another, medieval Christians paying attention learned that the Islamic sacred book pulled Mary into its own tradition.

This point occupies one paragraph in one of the Dormi secure’s longest sermons. Preaches cut and added what they chose. Surely they would never pick such a dangerous example, right?

Franciscan friar Marquard von Lindau’s The Ten Commandments was one of the most popular books among German lay readers. It’s a moral treatise with advice for widows, avoiding anger, dealing with lust. Marquard does not have much time to soapbox for Immaculate Conception. The argument he picks? Muhammad’s quotes from the “liber alcoran.”

Right out in the open. Available firsthand to laity with no hesitation. And apparently considered a pretty powerful argument in favor. Yes, Marquard’s actual readership was small. But it shows that clergy absolutely used hadith to support the Immaculate Conception when talking to lay people—affirming a connection between Christianity and Islam.