What happened to my plug adapter?
What happened? Most likely bad contact between pin of your plug and metal inside the wall socket. It is mechanical issue, deformed socket, tolerances which met on the wrong side etc. There was either high resistance contact or small gap between your plug and the socket, leading to arcing (that's what you heard), heating up the socket further and melting the plastic.

Why did the 13A fuse on the microwave plug not blow? Because the current flowing through was not much higher than 13 A (it was probably significantly less than 13 A in fact). The current was still limited by the connected microwave oven, there was no short circuit.

But one does not need 3 kW (230 Volt times 13 Amps) to melt small piece of plastic. Normally, there is almost zero voltage in the socket, so almost no dissipated power even for high currents. But because problem described above, in your case there was significant voltage drop and therefore power and heat melting your socket's pin.