But the variable they used shows a statistically significant influence.

They could have come up with a variable that is "the first letter of the first name", but then, the result will not show such strong difference between the groups. In other words: if a variable shows strong difference, then this variable is a good choice, and there is a real mechanism below this variable, and classifying people according to this variable corresponds to separating people based on the fact that they, for real, are really different groups that go through really different mechanism. (it does not mean causality, but even in case of confounding variable, it is still a pragmatically good choice of classification)

Then, the question is what are these mechanisms. There are a bunch of hypotheses, for example:

1) society has rewarded people with "efficient gene" to be in high class, so people in high class are naturally better. This is the most naive hypothesis, and it corresponds to a common bias and a lot of studies or observations indicates it is probably at least a very small reality (adopted children, "imposters", ...).

2) higher class family have values and standards that educate their people to try to be better at their jobs. Again, it's a common bias to believe that, but there is no real proof of it (the fact that some people think it's "obvious" is in fact the proof that they are biased).

3) higher class family have access to better school education because of resources. It is unfair, and in a modern society, we should want to compensate for that: everyone should be born with the same opportunity.

4) higher class family have values and standards that educate their people to try to consider themselves as better as the others, to present their skills less honestly and to have less guilt of take the position instead of a more qualified person. Inversely, lower class family have values and standards that educate their people to be more modest than what they really are. If you think that this hypothesis is bullshit but did not think the same as hypothesis 2, then, you are clearly biased: this one is the same under-lying idea (different education), but you prefer the one positive for the higher class.

5) higher class family have better contact and better arguments to get better salary. It is unfair, and in a modern society, we should want to compensate for that: better skilled people should get the job.

6) higher class family have better reputation to get better salary. Similarly to a lot of studies that shows (using blind approaches and scientific methodology) that for two candidates with same skills, the recruiter is unconsciously biased to choose the one that correspond to the caricatural mental picture of who would be an ideal recruitee, recruiters tend to be unconsciously push to believe a candidate would be better if they know this candidate has characteristics that are in fact uncorrelated, such as "coming from this fancy school", "being of higher class", ... I personally think it's difficult to believe this does not has a big impact, because such impact has been observed in plenty of studies in a slightly different context (gender or race, for example). Also, it explains why the "imposters" works as it does ("imposters" = people from working class that have pretended to be higher class)

7) ...

The variable does not imply anything on those mechanisms, but it does not need it to still separate into groups that are, indeed, subject to different treatment.