Social mobility and income inequality
The argument is correct

Social Mobility and Income Inequality are simply two distinct phenomenons. Mobility is arguably the more complex one, and there are many different ways to define it. As an example, absolute mobility refers to someone earning more income in absolute terms (\$ per year), while relative mobility refers to someone earning more in relative terms (going from the 60th to the 70th percentile of the income distribution). Note that the latter could occur for constant personal income, if income of all others decline. Another major distinction is made between inter- and intra- generational mobility. I would attribute part of the reason mobility is not featured often in political discource to the ambigiuous meaning of the concept. Related to this, there is no generally accepted single indicator (like the Gini coefficient for income inequality) by which mobility could be measured.

The second reason is that measuring mobility requires repeated observations of income distributions for the same individuals, ideally over a long period of time. These kind of data are still rather rare and not available for all countries in the required quality.

The scenario you have in mind ("the guy is not poor because he is likely going to earn a lot") is essentially a prospective task where one would estimate future earnings paths. While this can be done in principle, the precision of such studies are far below e.g. a detailed analysis on poverty ratios for various demographic groups.

Summing up, measuring income inequality ignores the movement of individuals between two points in time, which should be kept in mind when talking about income inequality. Povery is much more a problem if it's not a transitory, but a permanent phenomenon from an individual perspective.