I know that there are exceptions to this (My GF had elementary school that was K-6 for instance), but the three groups of grades K-5, 6-8, and 9-12, at least in the United States, seems to be by far the accepted standard anywhere I've been.
How did this differentiation come to be and for what reasons, and how did it manage to take hold so nearly universally compared to alternative breakdowns that were considered?
I've written a few different answers to this question under my old handle (u/UrAccountabilibuddy) so I'll be pulling from them a bit. Generally speaking, when we talk about this kind of history, we're talking about what happened in the Northeast of the country. As public education spread west and then south following the Civil War, most of the teachers and school administrators came from the east meaning they basically set up structures like what they had back home. So! From the top!
Virtually all countries with formal education have a three-level system:
This particular model can be traced back to Greek and Roman structures of education that moved young men through different settings as they aged. A German educational reformer in the 1630's, Johan Amos Comenius, was inspired by writing about the grammaticus and advocated for a similar system. This eventually evolved into the Prussian model, which informed the UK and America. The evidence of his reform advocacy can be seen today in the name of the Boston Latin School, America's first (1635) high school. Comenius' use of Latin as a synonym for "secondary school" referred to the common curriculum, known as a "classical" or "Latin", for older students of his era. It was this naming convention Boston fathers with means used when creating a place for their sons to prepare for Harvard (which, incidentally, meant pretty much any white boy/man who wanted to attend, regardless of age.)
In the decades after the Revolutionary War, especially in rural areas, formal education was primarily available to children of families with access to funds to hire a schoolteacher or tutor. There were some collectively-funded schools, typically called Academies, but not many. By the 1830s, though, the concept of publicly-funded schools for all American children as a net good for the country caught on among American adults in positions of power and gave rise to the common school movement. Advocates pushed for rural communities to fund a schoolhouse and school teacher and serve the community's children. (Cities on the east coast were already headed in that direction and developed their own systems.) They also pushed for the feminization of the profession - which not only resulted in young, unmarried white women rather than white men becoming the most common figure in front of the classroom, it created the concept of the schoolhouse as an extension of the home for both teacher and students.
Students generally attended school for 6-8 weeks in the summer and then 6-8 weeks in the winter - if they went, they went. If they didn't, they didn't. Functionally, this meant there was little attention paid to class size in one building until well after The Civil War because, generally speaking, if the population of a community expanded beyond what a single teacher or schoolhouse could handle, they would typically build a new schoolhouse. (At one point before consolidation in the 1940s and 50s, there were more than 100,000 schoolhouses in New York State. They would also just cram them into the existing school. The "Lancastrian Model" where older children were taught how to teach younger children was one short-lived attempt to solve the problem of overcrowding.) Urban schools where there were simply more children to educate were already working on different ways to solve the problem of over-crowding (among other issues), establishing levels of bureaucracy in schools, including the creation of a new class of educators, known as schoolmen. These men - mostly white
Before we get back to buildings, we need to talk about the early 1900's and what adults thought happened between children's ears. The idea that there is a transitional stage between childhood and adulthood - what we now think of adolescence - emerged as a concept in the late 1800's. The notion was championed in America by G. Stanley Hall, first president of the American Psychological Association and all together unpleasant man, even by the standards of his era. He firmly, and loudly, believed that children were irrational, illogical, and undeserving of respect as full human beings. Beyond that, he lectured and wrote about a sort of hierarchy of childhood. Children of color with disabilities were at the bottom, then white children with disabilities, then non-disabled girls of color, etc., with white boys as the top. He believed that children in the middle of this hierarchy required extensive work and patience to be prepared to join society and saw education as a way to accomplish this.
Part of his advocacy was to persuade schoolmen to separate children by race, gender, and age. A great deal of his advocacy fell on willing ears - separating by race was a no-brainer in a country that kept Black adults and children in slavery for centuries. Race-segregated schools were the norm until the late 1950s. Separation by gender made a certain degree of sense to educators but didn't catch on at a systemic level in the way that segregation by race did. (Evidence of weak gender-based segregation efforts can be seen on schools, especially high schools built as a result of the WPA projects during the New Deal era, with separate entrances for boys and girls.) Age, though, caught on in a way that we've had a hard time shaking.
Although Hall died in 1924, his work lived on in an increased educational, medical, and social focus on the the transition from child to adult - puberty and adolescents. It was his belief that something transformational happens when a child enters puberty and as such, they should be segregated from older and younger students. As schools began to merge into districts, this notion of extra time and space to transition into high school, rather than making a jump from grammar school to high school, seemed utterly reasonable and doable. This gave rise to the notion of an intermediate, or middle, school. This idea made its way into state policy and guidelines that shaped public education and while many states maintained the grammar school (K-8) structure, most adopted the idea of elementary, middle, high. The look of middle school varies by state and district but generally, they have fewer students than elementary schools and fewer class periods than high school. There's often an advisory period for social emotional issues and furniture that's neither small or adult-sized.
Middle schools moment to shine came in the 1950s with the arrival of The Baby Boom generation at school. The consolidation process was in full-swing, districts were created firm boundaries (typically shaped by racism and classism, laying the foundation of the concept of "good" school districts) and in many places, having three levels made staffing, transportation, and budgeting easier.