Labour is definitely the party of the affluent 'middle classes,' however much to claims its support lies in the 'working classes'
That says much the same as the article I linked to, M0nica, except that I don't think the LP does claim its support lies in the working class. It wants equality. Arguably, as this is a very unequal country, equality would benefit the poor more than the rich, but it is perfectly possible for people who don't stand to gain financially from it to want to live in a fairer society.
Political allegiance is not divided simply on class lines, as if it were the LP would have been in power far more often than it has. The phenomenon of the 'working class Tory' is well known. There are various theories to explain them. The Marxist view is that they have 'false class consciousness' ie they are people who feel a cut above their neighbours and think that voting Labour would somehow align them with those they look down on, so vote against their interests (measured objectively) in order to feed their sense of superiority. That point of view is summed up by the spoof Red Flag song, that starts 'The working class can kiss my ass/I've got the foreman's job at last'.
A psychological perspective is that they are people who prefer to see society as hierarchical with those 'at the top' being natural leaders 'born to rule', or that w/c Tory voters are those who like life to have rules with 'just because' explanations, so are less likely to question unfairness because 'that's just the way things are'. The 'deferential' working class, in other words.
I'm not sure what I think about those explanations, but to come back to the thread, a classic example of the first group would be (many of) those who bought council houses in the 80s. Thatcher knew that this would buy her support, and she was right. People immediately painted their front doors to show the world that their house was privately owned, and many shifted allegiance to the Tory party, as though having a mortgage instead of a rent book had changed their social class, even though their wages and work situation was exactly the same.
These days we have groups such as 'Mondeo Man', 'Disillusioned Suburbans' 'Red Wall Brexiteers' or the 'Patriotic Left', who cross outdated class lines and are categorised according to priorities and concerns.
What colour car do you have or did you used to drive?
US troops forced to act on the ground?
A drop in the ocean in the great schemes of things....but replicated by how many more
. The membership is predominantly from the educated middle classes (or was at the last election), and I think the same is true of supporters, although to a lesser degree*. What it stands for is fairness for all - 'the many, not the few', and it has never wanted to drag everyone down in order to achieve that. 