To work this out, it helps to think about what class actually means. Generally, this can be one of 4 completely different things.
- The government defines "social class" by occupation. If the main income-earner in your household is a professional, you are social class A. If the main income-earner in your household is an unskilled labourer, you are social class E. In a world in which plumbers (class C2) earn more than academics (class A), this is increasingly unconnected with income, but fits reasonably well with the ordinary English meaning of "class".
- Anthropologists (and snobs) define class by culture. The most famous anthropological study of English class distinctions is Nancy Mitford's work on "U" and "non-U" English More recently, Kate Fox's (highly readable - and recommended) book Watching the English deals with the issue in great detail. It is in this sense that America is a "classless society" - while there is a hereditary elite in America, they use the same vocabulary and drink the same beer as everyone else.
- Marxists define class by how someone relates to the means of production. Marx divided people into the proletariat, who made a living by selling their labour, and the property-owning bourgeoisie who lived off the profits of their business. To a Marxist, a city slicker on £200,000 can still be part of the proletariat if he does not own his own business. Will Hutton's analysis of Britain in terms of a 40-30-30 society is arguably an attempt to redefine Marxist class in a way that is relevant now that the middle class own the means of production through their pension funds - although Hutton doesn't use the word "class".
- Lazy people sometimes say "class" when they mean "How much money did your parents have?" Even if Wayne Rooney's children end up being total chavs, most people will not consider them "working class".
So what are people thinking about when they claim to be "working class"? It isn't point 1 - a manager who self-defines as working class presumably knows that he has a managerial job! It isn't point 3 either - nobody except the odd Marxist theorist uses the word "class" in this sense. Point 4 plays a part - saying "I am working class" implies "My parents were dirt poor", which in turn implies "Look at me and all my money - and I earned it all myself." But mostly it is about point 2.
Eliza Doolittle needed to acquire middle-class manners in order to get a middle-class job, because of the stigma that attached to Cockney culture in 1913. Nowadays, the tabloids insist that even a Prime Minister with a thoroughly upper-middle class upbringing engage in regular displays of vulgarity to prove that he is a "man of the people". So of course more people say that they are working class.