What is an analogue shift register?
There are a multitude of designs, but it transfers an analog charge level from one capacitor to another.  There is some noise accumulated in the transfer, but it is usually linear and/or proportional to the input.  The amplifiers used in the transfer circuitry introduce a small DC offset and the amplification is never exactly 1:1.  This means a single stage may for example add a few microvolts and boost the signal a little too much or not enough.  But these will be consistent from one operation to another, so its not random.  Minimizing these losses involves either increasing or decreasing the feedback resistance values.  Increasing them makes the circuit more power efficient but more susceptible to thermal noise and other effects, which are not necessarily truly random.  Decreasing them makes the circuit use more power.  Any stored analog signal will suffer from drift over time, but again, drift is not necessarily truly random either.