"When people insist that it is not a classless name, does this mean that people of, say, working classes can't use it"
No, there are no laws. 
Of course it would - or could - suit anyone. Anyone can - and should -use it. But names - like pretty much everything so many things in British society - are often inextricably associated with class. This is the point being made. Hermione has middle/upper middle class associations. It's not a classless name.
Once you know which supermarket a person shops in, which newspaper they read, where they buy their clothes, what accent they have, what sports the play/follow, what hobbies they indulge in, how they decorate their house, where they holiday - and this is before we even come down to their education, what they do for a living, and how much money/assets they have(!) - you get a pretty good idea of their class. Their name is just one small stand woven into a complex picture. Practically nothing in the UK is classless, when you think about it!
It is especially pronounced in the UK, but the US also has an ingrained class system, and even my small, Antipodean country is riddled with class indicators.