I don't know if other countries are as obsessed with class as we are; definitions are very fluid and I would say based more on education, thus increased earning power, rather than antecedents. It is the attempt to categorise people into deserving and undeserving workers that offends.
Very broadly speaking, upper class encompassed those descended from the aristocracy, usually with family wealth but supposedly with good breeding, (which I would not attempt to define.)
Middle class loosely described people with professions, formerly younger sons who had to earn a living, generally in the church, the law and the military and later expanding to include people with university qualifications and training
Working class was a euphemism for lower class, the 'rough poor' until nearly everyone had paid employment.
And 'working people' is still associated with working class which to many is a derogatory term, as is obvious from the response it provoked. Obviously, some workers are going to be taxed but not all, and the implication is it will be those who save, or as my father drummed into me, 'always have something in reserve,' which came from hard work and thrift and supporting his family through the depression, not a well-paid job.
Starmer made a foolish choice of words, and as has been pointed out, a barrister should know better.