The class system means social classes in society as a whole: Working class, middle class, upper class, mostly, with people staying within their class. In real life nowadays that translates pretty much in what happens in all countries: If you're rich and educated you tend to live in an area where people are also rich and educated and you interact socially with similar people most.
Despite being a monarchy with a nobility class the UK have actually historically been front runners in Europe when it comes to the rule of law and protection of individual rights, which is what matters above ideology. For example, is there still a "church tax" in Germany? Well even if the Queen is head of the Church of England that would be perceived as an unacceptable breach of individual freedom this side of the Channel.
Bottom line regarding the discussion at hands: none of this makes any practical difference to individuals and people thinking of moving to the UK.