Random, in its every-day meaning, simply means that uncertainty is involved. In the precise mathematical meaning it means that uncertainty is potentially involved. In any case, random does not imply any uniformity over the possible outcomes. Tossing a die with six sides numbered 1,2,3,4,5,6 produces a random event with equal probabilities for each outcome. Tossing a die with six sides numbers 1,1,1,1,1,6 produces a random event with possible outcomes 1 and 6, with unequal probabilities. Purely mathematically, tossing a die with six sides numbered 6,6,6,6,6,6 produces a random event with one outcome $6$, whose probability of occurring is $1$.
The assumption on birthdays is that each day of the year is as likely as any other, thus that the distribution is uniform.
It should be noted that the assumption of uniformity in face of uncertainty is a very common mistake. Not only it is wrong in many finite situations, in certain infinite situations it can easily be shown that no uniform distribution exists (this leads to some paradoxes, like one variant of the two-envelope paradox).