In my mind, people's class is largely about their upbringing, which influenced things like attitudes, education and income. Some people 'break out' from the norm of their background, perhaps in terms of education or income, which might or might not alter their attitudes and values. They may then seem a different class and their children might be brought up very differently and essentially be a different class.
Working class to me tends to mean no degree, or possibly a degree from a low ranked uni which doesn't tend to get people into the professions. Limited qualifications might result in the kind of jobs which have very fixed working hours where people start and end at a certain time and are largely directed in what they do. These can involve promotions and becoming supervisors etc. Income might vary from only what comes from benefits to really good incomes. In my mind many working class value stuff more than experiences so might like big TVs and regular new sofas and not have huge amounts of savings, so as a previous poster says, if something goes wrong,mothers isn't much of a buffer. There might be buying on credit - so cars on finance plans, technology on similar, often using Help to Buy etc for houses....things that mean you pay more innthenlomg run because you couldn't pay upfront. But some working class are very frugal too. Attitudes can be different and about avoiding the big struggles of huge hard work at school, but being focused more in happiness than academic and economic success....obviously generalising a lot here. Probably less planning ahead for the kids than the middle class do and a bit more of going with the flow or only realising eater late that to have certain options, choices needed to be made far in advance than they have been or were possible because of circumstances. So it's about having less choices.
In the past the vast majority of the population were working class. Few people had degrees and most people had low paid and low skilled work and were the norm. These days, lots of people don't want to be called working class but middle class. Because they own houses and have some qualifications and more money to spend they consider themselves middle class. Instead, the term working class has come to be used for the under-class - those who survive on benefits and which has negative con nations, such a s families in the drama Shameless. So perhaps there is this group who might have very little education and aspiration ( and of course not everyone on benefits is like this - the low aspiration is key to being under class) and then there are low paid workers, who consider themselves middle class because they have a lot of technology in their homes and other stuff...regular new clothes and stuff and big piles of Christmas presents....proving to the world and themesleves that they can't be working class.
Middle classes have more choices and less to prove. It's about attitude and being confident and this confidence often comes from knowing you have a background, or an education, or money in the bank, which means you will have options. Even if things go a bit wrong, you have resources to help you out. These might arise from family who will bail you out, from education which means you are resourceful and can get a job, or money in the bank to draw on. There's no need to prove yourself with lots of stuff because you already have that confidence.
Often those who grew up working class and became affluent still feel a need to prove themselves. Having plenty of everything still has novelty factor and remembering soft the feelings of others knowing you lacked certain things, can lead to the desire to show you have plenty of everything.....is this why people buy vast piles of stuff for their kids at Christmas ..because they didn't have it themselves? This can still be prevalent amongst those with high incomes and who are well educated. As time goes on, they might 'learn' to value experiences more than stuff and to teach their kids some of those values and because their kids grow up with those values, they essentially are middle class.
I think our backgrounds always influence us, even if we like to think we've moved on. They influence our attitudes and values to some degrees. From outside people often just see the level of affluence and not so quickly attitudes which prevail.
The affluent adult who is very keen to promote education and aspiration in their own children, might have had a bit of a struggle to get where they are and want to make sure their children earn the way to avoiding slipping back down. A different affluent adult who is more laid back about driving towards a traditional middle class career might be someone from a working class background who still holds those views or someone from a more affluent upbringing themselves, who because they have always known security, aren't so worried about their children's future because they take affluence as a given and just expect it to continue.