No, that's not the point. A society with almost no income inequality, where everyone makes almost exactly the same amount, would have nearly perfect "mobility" as measured by the correlation between the percentile of parent and child earnings (a few random dollars would move you a lot!). But it wouldn't be a "mobile" society in the sense we usually mean, where the poor can become rich (and vice versa) because there would be no poor and no rich!
It depends on what you're trying to measure by "economic mobility" though. I can think of valid reasons to want to know the percentile change between a person and their parents, irrespective of the absolute value difference, and reasons to be interested in the different-but-related measure of absolute change in income/quality of life.
It's generally better to start with the question "what do I want to know?" and then looking for a suitable statistical framework that will answer that question, rather than starting with a ready-made statistical analysis and then trying to use it to answer arbitrary questions.
I can think of valid reasons to want to know the percentile change between a person and their parents, irrespective of the absolute value difference, and reasons to be interested in the different-but-related measure of absolute change in income/quality of life.
Sure, but when comparing nation A to nation B, it's generally more interesting to look at either the absolute dollar difference or some relative quality-of-life measure (you could call the latter "class", I guess). Folks moving around within a tightly distributed income distribution don't really gain or lose much of either, relative to someone in a society with larger disparities.
Mobility is about moving between classes of society. So, the society you just described would be rightfully described as having high social mobility. What it probably does not have is high average wealth.
So, the society you just described would be rightfully described as having high social mobility.
But it would just be economic Brownian motion. Social mobility isn't (I hope!) valued because it shows that a society produces random outcomes, but rather that it enables people to achieve to the limit of their talents.
Social mobility is defined somehow and that somehow is not "the absolute total measure of fairness of any society". And speaking about virtually any country in the world, including North Korea, they all have lower class and upper class. Your worry about totally classless society is purely theoretical.
If we would live in classless society, social mobility would be less important measure.
Plus, you define achieving the limit of the talent in purely financial form. If I am genius mathematician, achieving limits of the talent mean doing great (potentially useful) math somewhere at the university. That does not translate to neither "high class" nor "rich".
