Yes, and as always the systems are sufficiently different that analogies are always going to be imperfect. Indeed, the constituent colleges play a significant role for undergraduate students at both Oxford and Cambridge, further making analogies very difficult. I analogized them to Berkeley, but you could also analogize them to the Claremont Colleges . . . we just do things differently.
From a practical standpoint, though–the government sets the tuition for Oxbridge (and most UK universities), and it is currently 9,250 GBP for UK citizens, under $12K at today’s exchange rates. And there is also a government loan program for up to the full fee. So that of course is way less than full tuition at our nominally comparable private colleges.
For “overseas” Oxbridge students, it depends on the course, but typical humanities and social science courses at Oxford are currently 35,080 GBP (about $45K), sciences 44,240 GBP ($57K).
So that is why at least from that perspective, it is sort of like the in-state/out-of-state situation at our public universities, understanding the whole UK is “in-state” for Oxbridge.
And then, say, Scottish universities have an even lower rate for Scottish students (1820 GBP, about $2300), then the 9250 GBP for other UK students, then again higher fees for overseas students. Like, Edinburgh is 24,500 GBP (around $31.5K) for most humanities and social sciences, and 32,200 GBP (over $41K) for sciences.
By the way, Scottish undergraduate degrees are usually four years, English usually three (unless you add a masters), so total cost of degree is actually very similar between Edinburgh and Oxbridge for overseas students, but for non-Scottish UK students the English degrees are cheaper by 25% (although sometimes students can qualify for second-year entry to Scottish universities).
So you can see why Scottish universities seem particularly interested in marketing themselves to US applicants. And yet if your alternative would be full tuition at a selective private college, one can still see the appeal.