So, first things first. Yes. That is often said. It's also misleading and typically used as a rhetorical device by speakers or authors who are advocating for a particular change. I'm the author of this Wikipedia article where I lay out the problems with the phrase "factory model" or "industrial era" schools. I also answered a question about it here.

But to the second part of your question - the relationship between school and the workplace is a fairly new construct. Like 1970s new. Before then, if there were skills a young person needed to do a particular job, they would learn them on the job. To be sure, there were schools that had vocational education programs focused on particular fields or even clubs - Future Farmers of America, Future Teachers Club, Future Nurses of America, etc. all had their start in the first half of the century - but the notion that school was about preparing children for the workforce, in particular, is fairly new.

Instead, once the common school movement hit its stride in the 1830s or so, the push was for a liberal arts education. And for the next century, the debate was about the flavor of that liberal arts: classical or modern (AKA English). Both included basic literary and mathematics at younger grades but a classical liberal arts education has Greek and Latin at its core and is focused on thinking as the ultimate outcome, not specific knowledge. A modern curriculum, on the other hand, is more about practical knowledge - English literature, history, science, math, etc. A classical curriculum is about teaching children how to think about hard things and know the things that smart men know. The modern curriculum is about teaching children lots of things that will help them be informed citizens and members of society. By the 1940s or so, American public education had settled into the modern liberal arts curriculum.

What this means practically speaking is that there is no meaningful difference in the content of the education the children of a factory worker and a factory owner received - provided they both attending public schools and both were white. That said, there were marked differences in terms of the quality of the school itself, the count of teachers on the school faculty, the diversity of the student body, etc.

Neither the children of the poor nor the children of the wealthy were being trained to work in a factory. Instead, they were receiving an education that adults felt would prepare them to be good citizens and members of society. To be sure, this is not to say their educational experiences were identical for the simple reason that schools attended by wealthier parents were more likely to be newer. To have more funds for teachers (meaning lower class sizes) and more funds for resources (meaning more textbooks, etc.). There's also a bunch to be said about how wealthier parents have influenced the public school curriculum but that's also a more modern phenomenon.

All of that said, the secular private school system has long been willing to provide what wealthy parents want for their children. So, as public schools moved away from the classical curriculum towards the modern curriculum, private schools were happy to remain with the classical curriculum if that's what tuition-paying parents wanted.