I'm not a historian of education and don’t know much about the historical development of the house system in British schools, but I can speak to the literary history that Harry Potter comes from, so hopefully this answer will be helpful.
One of Harry Potter's less-known literary antecedents is the school story. This is a genre of fiction, aimed at children, which focuses on the child's education. I use the term "education" rather broadly here, as this means not just reading, writing, and arithmetic, but moral education. School stories are about the shaping of children into moral adults.
Traditionally, the school story is set in an English public school (what we Americans would call a private school—I know that’s confusing). The English public school was at the time (eighteenth and nineteenth century) a single-sex boarding school, run by a headmaster or headmistress. These schools did (and still do) have houses as a means of dividing up students to determine where they lived, what teams they competed on, and so on and so forth. These conventions were reflected in the fiction set in these schools.
The first school story is generally identified as Sarah Fielding's The Governess, or the Little Female Academy (1749), which reflected the growing concern with how children (of both genders) are to be educated. This interest only increased as time went on and more children were being educated outside the home, and school stories were a popular genre.
The exemplar of the school story is probably Thomas Hughes's Tom Brown's School Days (1857). Not many people take the time to read Tom Brown's School Days in 2018, but if you do, you’ll find a lot familiar about it. Tom Brown, a student at Rugby School, is part of a house (like Harry being in Gryffindor), plays sports (he plays rugby, Harry plays Quidditch), and has a wise headmaster. Thomas Arnold, Tom Brown's headmaster, is based on the historical Thomas Arnold, who was headmaster at Rugby in the 1830s when Thomas Hughes went to school there. Arnold established a series of reforms at Rugby which were eventually adopted by other schools, including the praepostor/perfect system. Though in terms of personality, Arnold and Dumbledore are not much alike, they share similar roles in their respective novels as the wise guardians of their school who help guide the protagonist.
Tom Brown is quite a moralizing book, in ways which are out of taste for a lot of contemporary readers. The text reflects Hughes's belief that the school shapes the boy into an upright moral man who will serve his nation. If you’ve ever heard the quote that "X Battle was won on the playing fields of Eton," this also reflects the same ideas about what education is supposed to do. (Side note: this quote is often misquoted and misattributed; it’s traditionally supposed to have been said by the Duke of Wellington about Waterloo, though I’ve often heard it attributed to Winston Churchill as well. Neither one said it, but it is telling about the attitudes of the time.) Harry, of course, does not learn to be a pious Victorian at Hogwarts, because Rowling is writing to a different audience and different values. At Hogwarts he does learn to be brave and equitable, in keeping with the themes of Harry Potter.
Tom Brown was widely read by English children for more than a century and influenced future generations, who wrote books about children learning to be their best selves at boarding schools. The Harry Potter series fits into this tradition both via its setting and how that setting is used. Hogwarts, like Rugby for Tom Brown, is where Harry becomes who he’s supposed to be. The house system is a frequent feature of the genre because that’s how schools tend to be organized in Britain.
If you’re interested in Harry Potter and the school story genre, I recommend David Steege's "Harry Potter, Tom Brown, and the British School Story," which is the definitive piece on that aspect of Harry Potter.
Just as an aside, I’m aware of a US public middle school that’s used “houses” since 1957. It would have had the same structure as Hogwarts. We had separate dining halls, separate core classes and separate administration (like house heads in Harry Potter). Shared library and gym etc.
No real competitions though. This is because it was funneled from 5 different elementary schools and then mixed up in 6th grade randomly. So roughly 2/3 of your friends were in other houses. Sometime in the last 25 years they have switched to 4 houses from the previous 3.