Both the circuits might be producing equal voltages. But the looking in DC incremental impedance should be different for both circuits. In fact the first one would have 2/gm and the second one would have 1/gm. This means that if we had a capacitance in the output node, the first one would halve half the bandwidth of the second one. Probably we can see the 2x current in the second one as a reason to get this extra bandwidth. In Bandgap you might be more worried about the feedback amplifier's poles than this pole though.

An emitter follower shifts the voltage up. Do you agree? A resistor shifts the voltage up. Do you agree? Do I now tell both are doing the same job hence both are equivalent? The diode connected transistor is equivalently a resistance of 1/gm.