How did Fechner justify the assumption that the just-noticeable-difference in sensation is constant? As stated here on Wikipedia: 1

Weber's law states that the-just-noticeable-difference (JND) of an intensity of a stimuli divided by the intensity of that stimuli is always constant. 

Mathematically: $\Delta(I)/I=Constant$

... where $I$ here means the physical intensity of sound, light and so on.  $\Delta(I)$ is the-just-noticeable-difference.

Then came Fechner and assumed that the-just-noticeable-difference "in sensation" of a stimuli is constant as well, hence: $\Delta(I)/I = Constant = \Delta(S)$

$\Delta(S)$ stands for the-just-noticeable-difference "in sensation."

On what basis was Fechner justified in assuming that the-just-noticeable-difference in sensation is constant?
It was not an assumption; it was a decision. It allowed for better quantification, it gave an operational definition in increase in sensation, which allowed for more common ground in research between different labs. In a way, it's a definition by pure counting: when the subject feels a change in sensation, add one. Then this absolute scale is plotted against the ratio scale of increase in stimulation.