How does the voltage vs current characteristics of this VBE multiplier compare to those of a simple diode connected transistor?
The best way to answer at this question is do a simulation/hardware setup. Consider the following circuit representing the VBE multiplier circuit:
Figure 1. VBE Multiplier
R4 and R5 represent the potentiometer, and the simulation will use 3 values by configuring the x value to be 1kΩ, 2kΩ, 3kΩ.
The simulation representing the voltage vs current characteristic for the multiplier is presented in the plot below, where the y-axis is the current (uA instead of uV) and on the x-axis is the voltage
Following the plot, we can observe that the VBE value is dependent on the potentiometer value: for smaller resistance values (R5) the VBE is larger.