Where is the fallacy in this issue related to electric charge quantization? Suppose there are two identical conducting objects; one with 5 units of positive charge and the other is uncharged. Initially these are separated and later these are brought into contact. I guess net charge will get distributed equally between the two objects. That implies each body will have 2.5 units of electric charge which contradicts charge quantization since if I continue sharing charge like this, eventually I will have a conducting body with fractional charge. Where is the mistake in my argument?
 A: If you brought two identical and perfectly conducting balls together and there were and odd number of excess electrons then one electron can sit at the point of contact while the remaining even number can have half be on each conductor.
So each conductor has an equal number of charges. If you then symmetrically pull the spheres away from each other then that one charge that was originally shared between the two (sitting at the point of contact) feels zero net force towards either ball (it feels an equal repulsion from each sphere, so zero net force), so it stays in between the two balls.
Now in reality your spheres won't be perfect and won't be equal and there will be thermal motion so it actually won't divide equally. But if everything were perfect and at absolute zero, you still wouldn't violate quantization of charge, as shown above.
A: You "guess net charge will get distributed equally between the two objects", but that guess is wrong. If you touch the objects and then separate them again, one might wind up with charge 3 and the other 2, or 4 and 1, or 1000 and -995, or whatever.
A: You have been misled, I think, by the term "unit of charge". This can have two different meanings depending on context. Three, actually. At the microscopic (but not the quantum) level, the unit of charge is the charge on an electron or a proton. If you interpret the question as implying this unit of charge, then your question is quite reasonable, and your objection perfectly correct. Yes, the charge will get distributed equally - as equally as possible. With quantized charge, the quantization will prevent a perfectly equal distribution, and in this case will be 3 electrons on one sphere and two on the other.
However, there is another meaning of "unit of charge", in which unit is used in the sense of SI or Imperial units. In this context, the unit of charge is the Coulomb, which is the charge contained by $$C = 6.24 \times 10^{18} \text{ electrons}$$  I don't think I've ever seen C evaluated to the full 19 digits, so I can't be sure if it is even or odd. As a result, it's impossible to tell whether (for instance) 5 coulombs of charge would divide equally between two identical spheres, but a disparity of 1 electron would be entirely unnoticeable when compared to the 1.56 x $10^{19}$ electrons on the sphere.
Finally, a "unit of charge" in this case is not really 1 coulomb. It is simply some arbitrary number of electrons or protons, and it is pretty clear that the question was asked in this context. One point of the exercise is to get you accustomed to the fact that the absolute number of charges is pretty much irrelevant to the question of charge distribution. 
Well, unless you start talking about 5 electrons, of course. 
A: Consider charge as particles that are free to move on the surface of conducting objects, are mutually repelled and generally want to get as far from each other as possible. It's better to work with negative charge, as those particles are then electrons and that's how it works in reality. For positive charge you track "holes" in the mass of electrons.
When splitting objects, you cannot make two perfectly identical parts so some subtle difference will always decide where to put one extra particle. If we still suppose you could make identical parts, then the situation is outside the scope of our simplification of charges as point particles and you cannot predict the outcome using this model. You would have to use quantum physics and model charge particles in more detail, and then it will be random quantum fluctuation, that will decide, which part gets the extra particle.
The "outside of scope" is just like in theory, you could balance infinitely thin needle on its tip, but in reality, infinitely thin needles do not exist and even if they did, you would not see them balanced as every time some random fluctuation will kick them out of the instable equilibrium and they fall over.
