Nearly every explanation of oil drop experiment I ever found concludes along the lines of
The charge $q$ on the droplets were thus measured, and were all found to be integral multiple of $1.6 \cdot 10^{-19}\,\rm C$.
How exactly do you get this random number? I mean, I could say, they were all integral multiples of $0.8 \cdot 10^{-19}$, and I'll still be right since $0.8 = 1.6/2$. In fact, I can just divide $1.6$ by any number $n$, and conclude that it is the fundamental unit of charge since the charge on the droplets is a multiple of it.
Until and unless we are certain that a drop gained exactly one electron and measure it, there is no way to conclude the value of $e$, isn't it?
Best Answer
Millikan's oil drop experiment first and foremost serves to establish that electron charge is quantized. However, as you say in and of itself it does not exclusively specify the charge $e$, as it could be $e/2$ as you mention. However, the simplest interpretation of the data is to say that the charges are quantized with charge $e$; if it were $e/2$ instead, you'd have to tell me why should I expect every droplet to have an even number of electrons. Essentially, you'd need some kind of conspiracy going on, some extra physics to observe such phenomena.
However, there are also other ways to measure $e$, such as measuring the shot noise in a current-carrying wire. Such experiments give us even more confidence that the electron charge is $e$. In fact, in the fractional quantum hall effect, if one tries to do the same shot-noise measurement to experimentally verify $e$, one gets a rational fraction such as $e/3$ instead. This phenomenon actually led to the Nobel Prize in 1998 because of the non-trivial physics that was discovered. All in all, because of this and more, the value of $e$ is more or less an established fact (the evidence is more than the oil-drop experiment).