As air is accelerated over the wing, pressure drops and so does temperature, which is reduced to dew point and finally a cloud forms, which can be seen on humid days. My question is, can this cloud formed over the wing of an airplane produce lightning activity just like some other kinds of "normal clouds"?
Lightning occurs when an electrical charge is built up within a cloud, due to static electricity generated by supercooled water droplets colliding with ice crystals near the freezing level. When a large enough charge is built up, a large discharge will occur and can be seen as lightning (from Wikipedia).
So you need two things for lightning to occur- a charge buildup in a place and a charge gradient (difference in charge, like between two clouds or cloud to surface) for lightning to occur.
Based on this, there is little chance of any charge buildup and discharge in the condensation over wings- it is too short lived for any charge buildup within itself and the charges over the aircraft wings are dissipated by the static wicks in the trailing edges- so no chance of 'cloud' to wing surface either.
Friction creates static electricity
On the principle you're right: Electrons are separated from air or water vapor molecules due to friction occurring in the boundary layer and other friction areas. Instantaneously there are two types of static electricity created: Negative where some ripped off electrons are located and positive where air molecules with a missing electron are located (these molecules are named "ions" to indicate their electronic unbalance).
Ions are naturally neutralized
However opposite charges are attracted to each other, the ions recombine soon with their separated electrons behind the aircraft, the number of isolated ions or electrons –the "electric potential"– at any time doesn't build up so that a difference of potential with the ground, another mass, or between ions and electrons is sufficient to create any significant electrostatic discharge.
Note the recombination is accelerated by the water in the air, as water –with impurities– is a better conductor of electricity than dry air. Condensation doesn't help for static charge creation on aircraft.
Huge potential differences are required for lightning
The difference of potential is measured in volts*. It requires 3,000 V to initiate an electrostatic breakdown for each millimeter of dry air –the voltage used in an automotive sparking plug is about 10 times this value to ensure a good spark– or 3 GV per km.
If the electrons were trapped on the aircraft surface, the potential of the aircraft would reach the value required to create lightning with the ground :-)
So, why lightning happens with clouds but not with aircraft?
Friction surface from wings is ridiculously small compared to friction surface in the huge cumulonimbus cloud. Static electricity created is not of the same order of magnitude. That's a first reason, but there is also a significant difference in how charges behave.
In natural lightning the electrons and the positive ions are created by the violent wind inside specific clouds. However, contrary to what happens with an aircraft, and for reasons not yet really understood, charged particles are then physically separated: Positive ions move at the top of the cloud while negative charges move at the center where they combine temporary with neutral molecules to form negative ions.
A smaller quantity of positive ions also moves at the bottom. This positive cloud bottom also moves negative charges of the ground at its nadir, which in turn reduces the thickness of the air layer to move through, and at the same time increases the electrical field gradient.
As friction continues in the cloud, the electrical potential increases until the difference of potential between the bottom of the cloud and the ground reaches the voltage threshold required for breakdown. The huge avalanche breakdown heat creates plasma used by particles in excess to join the attractive ground. This phenomena is complex and still not entirely explained either.
*: The volt is the electrical potential required to create one joule of work with a charge of one coulomb. See this nice page for more details.
can this cloud formed over the wing of an airplane produce lightning activity just like some other kinds of "normal clouds"?
No. Lightning requires large, long-lived clouds in which warm, wet air can rise, condense and fall again. That kind of structure can't exist in a cloud so small and temporary as a contrail.
Wikipedia's page on thunderstorms contains lots of information about what's needed.
What you see above the wings is condensation, but not technically a cloud.
Clouds are suspended in the atmosphere due to prolonged saturation. Not a localized momentary pressure drop.
Those condensations dissipate as quickly as they form, there isn't enough time for them to be charged.
TL-DR: lightning is made possible not by the availability of free electrons (a very common occurrence, in Nature and in the atmosphere in particular), but by the existence of some non-electric force which separates positive and negative ions (despite their electrostatic attraction), over scales large enough to give rise to a large, sudden increase in the number of free, energetic electrons, once a threshold value of the electric field is reached. No such forces exist over the surface of a metal.
All of the previous answers are correct, but they touch in passing (and not all of them, at this) on the key prerequisite for the formation of lightning: the maintenance of large-scale charge separation, i.e. on the scale of kilometers.
Ionizing gases is reasonably easy, and all sorts of forces are able the sever the links between the outermost (and least bound) electrons from their (aptly called) donor atoms. But this occurs rather uniformly across an average gas like the Earth's atmosphere, which means that, in any given microscopic volume, there are as many positive as negative ions. The ensuing electric field is nearly vanishing, and achieving the large electric field necessary for a lightning discharge (roughly 3GV per km, as pointed out by mins) is impossible.
For instance, on an airplane wing, which is an excellent conductor, electric forces will tend to keep the positive and the negative (=electrons) ions very close to each other, and no large-scale electric field will ever be produced.
The key to explain lightning is that there are other forces which tend to pull electrons and positive ions apart, over very large distances. This has been stated already by mins (although I believe less clearly), and by David Rickerby. The fact that the nature of these forces is not clearly understood does not detract from the fact that this is what makes lightning possible.
It is easy to understand why such large electric fields are necessary: an electron hitting a neutral atom can ionize it, thus freeing one more electron which will go on to produce twice more free electrons as our initial situation where we only has an electron. But in ionizing a neutral atom, the original electron will lose energy, and might be unable to ionize an another atom when the next collision occurs, and/or it might produce a new electron which also does not have enough kinetic energy to ionize the next neutral it hits: thus no cascade, thus no increase in the number of electrons, thus no lightning.
Unlesss... the electric field is so strong that it replenishes the energy of the original electron (and of course also of its newly-formed brethren) of enough energy between collisions to ionize the next neutral as well. In other words, the electrons re-supply themselves of energy in between collisions, courtesy of the electric field, so as to start the process again. This explains the 3GV per km: the energy necessary to ionize a loosely bound electron is roughly 3eV, and the path between collisions in the atmosphere is roughly 0.0001cm. This implies that a single electron will acquire 3GeV over a km (thanks to the electric field), which corresponds to a potential drop of 3GV over 1 km, exactly mins' figure.
The above also explains the sudden discharge: when then electric field exceeds the above threshold value, the number of electrons doubles every collision: the typical distance in air between collisions is 0.0001 cm, so that, over the course of 1km, a good value for the cloud thickness, one single electron would produce a cascade of 2^10^9 electrons. This number is unrealistically large (the total number of free electrons produced is limited the finite amount of energy available, roughly that of a capacitor of size 1km and electric field 3GV per km), but it gives a good idea of how a lightning is suddenly released. This is in fact a cascade, a sudden vacuum breakdown. For lower values of the elctric field, the newly formed electrons will not have sufficient energy to ionize the next neutral they hit; for larger values they will, thus causing a sudden increase in their numbers.
Airplane produce lightning ? YES. The thunderstorm electric field is multiplied by the airplane and may trigger lightning.
Cloud formed over the wing of an airplane produce lightning activity just like some other kinds of "normal cloud"? NO. The charge it may hypotethically produce is order of magnitudine smaller if even created.