I'm best positioned to speak about American schools and feel comfortable saying if a teacher from 1820 - or even 1920 - was zapped into a classroom today, they would recognize it as a classroom (one or two adults, lots of children, books) but that would be about it. A fair amount be likely be baffling and disconcerting.

To a certain extent, education systems haven't fundamentally changed in centuries because the central premise - one or two adults passing knowledge deemed important by their society onto a group of children - has proven successful for generations, in societies around the world. Generally speaking. I'm the author of this Wikipedia article and I'll plagiarize myself a bit to focus on one particular "schools haven't changed argument."

"Factory model schools", "factory model education", or "industrial era schools" are ahistorical terms that emerged in the mid to late-20th century and are used by writers and speakers as a rhetorical device by those advocating a change to the American public education system. Generally speaking, when used, the terms are referencing characteristics of European education that emerged in the late 18th century and then in North America in the mid-19th century that include top-down management, outcomes designed to meet societal needs, age-based classrooms, the modern liberal arts curriculum, and a focus on producing results. The phrase is typically used in the context of discussing what the author has identified as negative aspects of public (or government-funded) schools. As an example, the "factory model of schools are 'designed to create docile subjects and factory workers'". The phrases are also used to incorrectly suggest the look of American education hasn't changed since the 19th century. Educational historians describe the phrase as misleading and an inaccurate representation of the development of American public education.

There are some pretty fundamental ways primary school has changed since the 1820s. The most pronounced change is likely related to the nature of content. Up until the late 1800s, there was no real demarcation to speak of between religious instruction and the content in common or tax-payer funded schools, especially when it came to reading instruction. The texts children would read were religious (almost always Protestant) and there impetus behind giving children a formal education was typically tied to religious goals (i.e. saving a child's soul.) There was no explicit breaking with religious texts, they just were slowly replaced by more secular texts and readers. The main exception to that was a fairly dramatic shift in cities, most notably New York City, that resulted in the rise of parallel Catholic school system. Outside the cities, though, schoolteachers made the slow and gradual shift from religious to secular texts. There's a whole bunch to be said about other changes at the high school level but those didn't necessarily impact primary education.

Another important difference is related to how children were grouped. With a few exceptions, before the 1900s and the widespread adoption of grade levels (4 and 5-year-olds attend Kindergarten, 6-year-olds attend First Grade, etc.) and groupings (primary, elementary, middle/intermediate/high) young children would often be in the same space as older children. This typically meant uncomfortable seats or desks for the littles and behavior expectations that could be difficult for them to meet. (Another big change? A small child attending school in 1820 was much more likely to have a man teacher and feel the back of their teacher's hand or switch for misbehaving than a child in 1920. The latter child was more likely to have an unmarried woman as a teacher and corporal punishment was much more frowned on.) As grade levels and groupings became the norm (which emerged from advocacy for those interested in the newly emerged field of child psychology and child study as well as population changes), the buildings children attended were increasingly designed around their bodies. This isn't to say early primary schools looked like modern day ones do, but rather, primary school increasingly became fundamentally different than grammar and secondary school. Generally speaking, children would show up when showed up, participate in the tasks assigned by their teacher (learning to read, write, colors, songs, some history, playing outside, some sciences, numbers - fairly similar to what children learn now, but again, that's not necessarily a bad thing.)

There would be more shifts related to primary school following World War II with the expansion of the suburbs and the arrival of the Baby Boom generation in schools. Compulsory education was the norm by then and most localities has solidified a school tax structure that was dependent on the count of children and the idea of "failing" or "holding back" a child became increasingly common. This was also when schools began to pay more attention to children's birthdays and set hard enrollment dates. In the 1960s and 70s, there was a push for increased school safety (most noticeable in changes in playground equipment) that shifted again the look primary school.

So, primary school has changed in big ways and small and remained the same in a few fundamental ways. When someone is making the claim school hasn't changed, odds are good their advocating for a particular passion of theirs.