Why does this unity gain transistor phase shifter not drop its amplitude with frequency?
My feeling however tells me R5 and C2 form a simple RC low pass filter
  and although fed by a symmetric input voltage tapped from the
  transistor's collector and emitter, I would still expect the output to
  drop at 6dB/oct from about 200Hz


At low frequencies R5 is dominant and the output is 180 deg phase shifted and unity gain (near enough) and at high frequencies C2 is dominant and produces a zero phase shift because the voltage feeding C2 is from the emitter.

There is no significant low pass filtering to consider here.

Consider that the voltage at the emitter is the same as the voltage at the base and call it Vin. The voltage at the collector is -Vin: - 







simulate this circuit &ndash; Schematic created using CircuitLab

Therefore the voltage at the midpoint of R and C (Vout) is: -

\$V_{IN} - \dfrac{2V_{IN} \cdot X_C}{X_C+R} = V_{IN}(1-\dfrac{2X_C}{R+X_C})\$

Therefore Vout/Vin = \$\dfrac{R-X_C}{R+X_C}\$

This is all without using complex numbers but if you did the analysis of top and bottom of the equation above, the magnitudes are equal: -



The above picture is when R and Xc are vectorially added together. Now if Xc was negative (as per the numerator in the equation), Xc would, of course point upwards BUT, the length of -Xc and R added together is exactly the same magnitude.