In the West at least, this is largely an outgrowth of patronymic naming systems in Europe. I have written in this sub about the history of "last names" several times before which will have more info and sources. One of the "last name" systems in use in Europe in the Middle Ages was called patronymics. Your father's first name became your last name. So if your name was John and your father's name was Peter, then you would be called John Peterson. The actual spelling varied from language to language in patronymic societies: in Dutch, German, and Scandinavian countries, it was written "-sen", in Polish it was written "-ski", in Russian it became "vich", etc. and then each had a similar system for girls: in Dutch it was just the one letter "-s", in Sweden it was "-dottir", in Poland it was "-ska", etc.

But it was also traditional bordering on mandatory (unless there was some serious family drama) for firstborn, second-born, and third-born sons and daughters to be named after their four grandparents and two parents, and if any of those kids died young, then the next-born kid would reuse the name.

John Peterson's firstborn son would be named Peter (after the child's paternal grandfather) so he would be Peter Johnson. And his first son would be named John Peterson. And so on, in an infinite loop. While the third born sons would be Peter Peterson and John Johnson. Consequently, because of this tradition, the pool of names in patronymic societies tended to be very small, and so everybody had a second "last name" too.

These were called "cognomen" which were actually based on the older system of "bynames". They were basically just the person's occupation or location or physically description, and look a lot like many last names today. The only differences were that these were a second last name and not the only last name, and we're not strictly inherited but could change generationally. So John Peterson the Smith might have a son named Peter Johnson on the Hill.

Strictly speaking, though, this second last name was used a bit more like "Jr." and "Sr." or like a middle name, and wasn't always mentioned unless two people were going to be confused. Everybody would know John Peterson the Smith as simply "John Peterson" but when he needed to be differentiated from John Peterson the Black who lived a few miles away when someone asked "Which John Peterson are you talking about?", then the cognomen would be mentioned.

In England and France, though, they never really adopted a true universal patronymic naming system. The jumper straight from the older "byname" system where last names could change generationally straight to the modern "fixed surname" system where the last name stays the same for generations to come. Sometimes a patronymic might be used as a byname and then as a surname, but the Continental patronymic system was never really adopted in either place.

In England and France, for a very long time it was not traditional for children to be given a middle name. In fact, in England, this has only really come about since after WWII and is a result of American influence. For instance, two of the Beatles had middle names, and two of them did not.

From the 1500s through the 1900s, the patronymic systems in Europe were all dropped at various times in favor of the fixed surname system, until Iceland remains the only "European" country to continue to use patronymics. This was sometimes done voluntarily, but often done through force by one country taking over another, or one pro-surname monarch taking over from a pro-patronymic monarch.

However, when the patronymic systems were dropped, often by force, the people did not necessarily drop the first name naming custom of naming children after grandparents and parents. And this is why middle names became traditions in places like Germany, the Netherlands, and America, but not in England.

In America, the English took over New York from the Dutch in 1664, and forced the Dutch population to adopt "English names" of fixed surnames in the 1680s. And they weren't allowed to use a name that someone else was already using, so the "Johnson"s and "Peterson"s and "Williamson"s were pretty much off dibs. Most families, then, took their patronymic last names as their new fixed last name, which is why all those "Van" names appeared ("Van Buren" was a family from the town of Buren in the Netherlands, same with "Van Patten", "Van der Bilt", etc.). But other names like "White" and "Smith", etc. were used too.

With the first name naming convention still in place, this eventually caused a problem, but it was a problem that didn't appear for a couple generations.

In the 1680s, Jan Jansen de Smidt becomes the first in his family to have an "English name" and becomes known as John Smith. John Smith has ten kids including four sons in the 1710s and 1720s. All good so far. Those four sons grow up and they have sons of their own in the 1740s and 1750s. And they each have a first born son named after the paternal grandfather. Now there's four different John Smiths, all cousins, all around the same age, all living in the same county, or even in the same town.

When they grow up and need to pay their property taxes, the tax officer needs a way to differentiate them. How do you do it? Simple. They resurrect a modified version of the old patronymic system. The John Smith who is the son of Robert becomes known as John Robert Smith or John R. Smith. The John Smith whose dad is William is John William Smith or John W. Smith. And the John who is the son of John is John John Smith or John J. Smith.

Thus, the first time that middle names as we know them started to appear with frequency in colonial America is during the Revolutionary War in New York and other "Middle Colonies" in towns with predominantly Dutch or German populations. Ten different John Smiths are listed on the militia rolls and are all owed a salary and later a military pension, so they start using middle names and middle initials to get their due.

The same thing happened in Germany, the Netherlands, and other places in Europe under similar patronymic-to-fixed-surname transitional circumstances. So not only did they start to appear natively in America in the 1700s, they also began appearing in the 1600s-1800s in parts of Europe, too.

By the mid-1800s, this tradition had spread beyond just Dutch and German communities in America as those populations spread out to the frontier and into more mixed communities, and intermarried with spouses of other cultural backgrounds. And the tradition of the middle name just being dad's name began to change to being whatever the parents liked, just as first names began to change, too. It was a bit of cross-culturalism because the English weren't so strict on the first name business, especially among the Puritans (who often had quite oddball first names).

Thus, John William Smith may or may not name his son "William" after the child's grandfather because he's no longer part of the Dutch Reformed Church but his wife was brought up Presbyterian and the two left their churches to get married in the new-fangled Methodist church. They name him Matthew Jacob Smith because they like the sound of it.

Again, similar situations came up in parts of Europe, but for people who were culturally English or French, this was not quite as common until the last fifty years. Thus, it's not surprising that of the first seven presidents, only one of them (John Quincy Adams) had a middle name. Beginning with Martin Van Buren (culturally Dutch), all U.S. Presidents since have had a middle name. But in England, it's only become common to give a child a middle name relatively recently. Among older English people even today, it's not unusual for them not to have a middle name.

EDIT: On mobile, so sorry for any spelling mistakes.