Tough question. Even as a native speaker, I originally disagreed with the accepted answer. But the reason that it's correct becomes clear if you explicitly add the implied (but missing) "that" within the sentence.
In many cases, the clothes that people wear identify them as belonging to a particular social class.
This makes it clear that the sentence could be simplified (noun-verb-object) to:
The clothes identify them as belonging to a social class.
That is obviously referring to people, whereas if you used "themselves", you would end up with
The clothes identify themselves as belonging to a social class.
Which shows that the accepted answer is precisely correct.