It doesn't surprise me that people self identify as working class, because self identifying as middle class is seen by many people as being full of yourself.
^^this. If you think about the USA, the vast, vast majority of people describe themselves as 'middle class' whereas in Britain a very large proportion of them would be considered absolutely working class. It's a difference in national character; in Britain it's seen to be boastful or undesirable to assert 'You're better than you are' while in the US they are so highly aspirational that everyone is a
Millionaire In Waiting
.
I am from neither place so this is from an outsiders perspective, but I can tell you that when I worked in the UK I was fascinated by the rigidity of the class system and how people instantly placed themselves and others within a hierarchy. Part of it was accent, part was where they were educated, part of it was birthplace. As an outsider (who couldnt possibly read all the subtle markers of difference, nor give a shiny shit about them) I was able to seamlessly move across social groups in a way that locals seemed to do far less. This was in London, in a very established institution.
All societies have hierarchies of course. I don't think it's unique to the UK at all, although the British class system is possibly more overt (still) because it was insitutionalised for such a long time and it's still got real life, political and social implications.