You're allowed to reuse the same sample for multiple variables, pollsters do it all the time, they'll phone one person and ask them 10 different questions.
The fact that you're measuring faces as opposed to political leanings does not affect the underlying statistical methods. Confounding variables only come into play when you're doing regressions, and/or trying to prove causation. Since they aren't attempting to explain why the faces look like this there aren't any confounding variables.
That pollsters do something does not make it valid! ;) Yes, they do ask dozens of questions of a small number of people. But this is also mathematically dubious.
If you do multiple random samples, the chance at least one is way off increases. If you have a 95% chance of being close to correct on your small sample, and you do 20 samples, it is quite likely you are wrong on at least 1 of them.
My claim is that (1) there is a very large number of relevant facial features for the goal of the survey (find the "average" face), and (2) that being wrong on even a small number of them is significant. My basis for both is a combination of intuition and that we know humans are extremely good at perceiving faces and subtle facial differences. There are parts of the brain that are apparently adapted specifically to facial recognition, and disorders where facial recognition is impaired but nothing else, for example.
The fact that you're measuring faces as opposed to political leanings does not affect the underlying statistical methods. Confounding variables only come into play when you're doing regressions, and/or trying to prove causation. Since they aren't attempting to explain why the faces look like this there aren't any confounding variables.