I would argue that there is a massive difference: University education is not compulsory, unlike primary and secondary education.
Compulsory is very relative. What kind of job most mortals can find today, without University degree? Then, even for the public and "free" primary schools there is a catch. A family which can afford to send their kids to Harvard they certainly live in such a neighbourhood that public schools are "more equal" than schools in poor neighbourhoods. So even when a family is sending their kids to "public" schools in a chic neighbourhood again those schools are of much better quality (also surrounding streets are cleaner etc) than those in poor areas.
Nowhere is more evident than in China. Public education is often the best (except >50,000$ a year top of the line private ones) and you must strictly send your kid around the area you are living /own something.
Thus the real estate prices skyrocket around good public schools as people pay literally millions for holes next to a good primary public schools to be eligible to send their kid there. So you can see rich people sending their kids in public schools where good standing people (but not rich) are obliged to pay for less quality private ones!!
In the end there are much more barriers than just the simplified public -private. Segregation in education will always be a thing either visible or not, because humans like to put barriers for other humans whenever and however they can.
Most UK Universities are not private institutions. It is true they charge fees (don't get me started on the rise introduced last decade ), but that is also the case with public Universities elsewhere in Europe, although in other places they are much more heavily subsidised by the taxpayer.
In France you can have high level studies not just for free, but they can subsidise part of it you if you are serious in what you are doing. Hell they can subsidise even if you are not that serious! In other countries you should pay for the books, get very cheap rent in the campus and there are no fees.