I've heard a lot of times that our modern school system it's actually a vestige from industrial times, and it was originally used for teaching kids to be good factory workers. It's this true? How much has it changed since the 18-19th century?

Thanks for the tag, /u/voyeur324! To paraphrase my earlier answer and to directly answer your question, no, it's not true. One thing that modern-day authors frequently do is apply a modern lens to texts written in the late 1800s and early 1900s. American society at the time was fascinated by the notion of efficiency in every aspect of life. The administrators, principals, professors, and politicians who made decisions about education frequently used analogies to describe what they envisioned for schools. This included progressive authors who advocated for the children in schools and conservative or classically-minded ones who focused on systems and society.

From the Wikipedia article I wrote on this topic:

The most prolific user of this analogy was Ellwood Patterson Cubberley. He saw the logical, methodical approach of scientific management as a way for public education to adapt to influxes of children entering the system and to ensure the best outcomes. Cubberley wrote numerous guides for school administrators as well as a history book and was one of the most widely read educational authors of the 1910s and 1920s. He frequently used the metaphor of school as a factory:

Our schools are, in a sense, factories, in which the raw products (children) are to be shaped and fashioned into products to meet the various demands of life. The specifications for manufacturing come from the demands of twentieth-century civilization, and it is the business of the school to build its pupils according to the specifications laid down.

To be sure, Cubberley didn't literally think of children as raw goods. He was fairly progressive-minded and advocated for school as a place that was safe, clean, and interesting for children. A few other ways we know it's not true is that children often made up a significant portion of the factory workforce before child labor laws. There would be no reason to educate children to work in factories where they were already working. Likewise, some factories, especially in NYC and Chicago, had schools on the factory floor. Children would attend class with a teacher for an hour or two before or after their shift and the teachers almost always focused on basic literacy, mathematics, and Americana (general history and routines).

The second part of your question - how have they changed - gets at the problem at the heart of the "factory model myth." In effect, when people claim schools were about creating factory workers, they're thinking about White children. When educators and politicians in the Northeast were advocating for tax-payer funded schools, Black children on plantations were working alongside their parents, presuming they hadn't been sold to a different owner. Same when Thomas Jefferson was advocating for a "common knowledge" in the late 1700s. He wasn't thinking about the children who worked on his plantation, nor the Indigenous children who'd been displaced from their homes. Until after the Civil War, it was illegal for enslaved children to attend school and in most cases, schools set up for free Black children in the North were poorly funded and resourced.

The most dramatic change to American schools can be seen in who makes up the student population and who makes up the teacher population. During the Colonial Era, most teachers were men, their students mostly the sons of men with access to power. As common school became the norm, women were recruited as they were cheaper and many believed, better suited to teaching than men. School became increasingly something girls did but rarely something a child with a disability could do. It wouldn't be until 1975 that public schools were prohibited from refusing to educate children with disabilities (or children they perceived as having a disability.)

Another key indicator that American schools do not prepare children for any one particular future is the nature of the curriculum. Almost without exception, American public school children experience a modern liberal arts curriculum - English, Math, Science, History, Physical Education, Art, Music, Foreign Language - for all 13 years. While there are places where children can enroll in specialty high school programs such as vocational education, it's up to the child and their parents. There is no state (as far as I'm aware) that requires children follow set paths as can be seen in German or Swiss schools. (There's a separate but related answers around "tracking" but that's it's own history.)