I'd agree, class has nothing to do with wealth really and people do get very confused. It's also blurred when it comes to values and behaviours, there used to be a lot more social mobility than there is now.
Someone who had family with doctors, lawyers etc going back the last 200 years would quite possibly be in a position of generational wealth far beyond, say a family that would be considered middle class where the first person who went to university went in the 1970s (or later). This said, I would say less jobs required a degree for entry (my grandparents didn't have them and we're in quite skilled jobs), where now it's easy to say you must have a degree or master's in X (librarian being an example).
Tradespeople will almost certainly consider them selves working class but will likely do very well financially if they have a good reputation.
I've got no idea what the average middle class salary or savings pot looks like, it's rarely discussed. People have different priorities and it will be different at different ages. But what I would say is that salaries that look decent on paper (say above 60k) no longer bring with them the lifestyle and ability to save that they would have done 5 years ago.
I find people 10 years older than me often have houses that were cheaper, have never paid a student loan and got their career established pre 2008 so we will never be comparable (and people 10 years younger than me will say the same about us no doubt).