Do you think this poem gives a true picture of our society today? Which details do or do not ring true?

You must remember that Auden was not describing, or satirizing, today's society, but the society of 1939 at the time of his writing the poem. Nevertheless, there are issues that were of concern to Auden then which remain pressing today—arguably far more so.
The first issue is the idea that people represent mere statistics to governments, with data processing being used as a means of keeping tabs on citizens. Auden is criticizing the tendency of governments to treat everybody according to how closely they adhere to societal norms; anyone who is not "happy" would surely have made this known to the government—except that the government equates happiness with being statistically normal and the same as everyone else.
Another issue is the idea that governments are always surveilling their citizens—this idea was just beginning when Auden wrote this poem, but is of far greater concern today.