How does the current mirror stabilize tail current in an op amp?

Question

Op amps often use the "slave" side of a current mirror to provide the tail current for the input transistor pair, such as in this simple op amp voltage follower schematic:

The "gates" node in the current mirror is constant. Therefore, the gate-source voltage of the two PMOS transistors is constant as shown in the first plot. This should result in a constant channel resistance of both M1 and M2.

Now the second graphic shows how the drain-source voltage over M2 varies by a tremendous amount when sweeping the common-mode voltage point.

But still, the current through M2 is constant (third plot). This demands that M2's resistance strongly varies, which seems to contradict its constant gate-source voltage.

Question 1: How is the current through M2 constant, despite a strongly varying drain-source voltage and constant gate-source bias?

Questions 2: When I replace the mirror transistors with PNPs, the current mirror performance becomes abysmal (see plot below). Why are BJTs so much worse in this regard?

Answer 1

But still, the current through M2 is constant (third plot). This demands that M2's resistance strongly varies, which seems to contradict its constant gate-source voltage.

It does not contradict anything, except possibly a false assumption that the mosfet is in triode. It is in fact not in triode, but rather in saturation where the drain current is well described by:

$$I_D = k ( V_{gs} - V_{th})^2 $$

and is fairly close to constant over a wide range of Vds voltages. There's a small variation from channel-length modulation which arises as a consequence of device semiconductor physics.

When I replace the mirror transistors with PNPs, the current mirror performance becomes abysmal (see plot below). Why are BJTs so much worse in this regard ?

Not clear from screenshot alone without also seeing the device models. In principle, BJT current mirrors are fine.

You may want to double check whether the transistor is in forward active mode and whether it is getting enough base current since -592 mV might mean insufficient gate drive (remember that MOS devices don't have gate current). Also try using a constant current source instead of a resistor on the input.