A magnetic pickup is made of magnets and a coil, or coils in humbicking designs. If you know your Faraday's law, you will know that as long as there is relative movement of the coil to the magnet, there will some induction of emf.
A change in the magnetic environment of a coil of wire will also cause a voltage (emf) to be induced in the coil. This change could be produced by changing the magnetic field strength, and can be caused by moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, vibrating a taut metal string close to the coil, or by simply tapping the coil/magnet.
Potting, i.e. engulfing the coil in wax, reduces vulnerability to vibrations caused by tapping , loud volume, or sympathetic resonance. These result in a loud tap, or feedback respectively.
A microphonic pickup is one where there is usually a thin diaphram, somewhere in the construction prone to vibrations. This could be a thin metallic pickup cover. Efffectively this vibrates and can transmit those vibrations to the coil/magnet and you may get an audible sound, amplified that you may recognise as a vocal sound if you shout into the pickups. Effectively these pickups are very primitive microphones. Sometimes, very thin hollow body tops offer this to the pickup, they pickup vibrations and transfer them to the pickup via sympathetic resonance.
Plastic covered pickups suffer from microphonic effects less than thin metal covered ones, though thin plastic covers allowed to freely vibrate can also induce EMF. Your typical modern pickup design, - say Bartolini, or EMG - are hermetically cast into resin, so the effects of feedback are minimised , but not eradicated completely.
All pickups are susceptible to tapping. If you tap a pickup and get no 'tapped signal' when you amplify the instrument - your pickup is dead. You can palm mute your strings and you would still get a signal even if you dampen the strings, so pickups can generate signals even with the slightest encouragement to induce EMF.
Your typical microphonic pickup which you can sing through cn be found on old Hofners, Teisco, Kent, Univox etc....