I think it kind of makes sense when one takes into consideration the nature of monarchies throughout history and the nature of democracies (in particular, countries that started out as democracies). In a county like the US everything the government owns is the property of the people, hence, government schools are "public". In a country like the UK, that hasn't always been the case, so a school that is owned by members of the public rather than the government, might be called a "public" school. That's my theory, anyway.