This question belongs more to philosophy rather than mathematics, so it might be out of scope of this site. But the general answer is that mathematical models are only an approximation to reality, and we choose those approximations which are convenient. For example, we can discover sometimes that the space/time is not really continuous but consists of some discrete objects. But this will not make calculus based on the concept of real number useless or obsolete. One can argue without end whether real numbers really correspond to something "real". Nevertheless they are useful in physics and engineering. (See, for example,
N. J. Wildberger, Real fish, real numbers, real jobs, The Mathematical Intelligencer, Volume 21, Issue 2, pp 4–7.)

Same with probability. Continuous distributions historically arise as approximations to discrete distributions
(normal distribution is an approximation of the binomial distribution via the de Moivre-Laplace theorem). But normal distribution is much nicer from the mathematical point of view and therefore it must be used, even if when the "real" distribution under consideration is binomial. 

In other words, the accepted axioms of Probability are chosen (from the several proposed systems) because of their mathematical convenience, rather than because they better approximate the "real world" from the philosophical point of view. 

(Of the systems of mathematical foundations of probability which were competing
with Kolmogorov's axioms, I can mention those proposed by S. Bernstein, R. von Mises and H. Steinhaus. And philosophic considerations played a secondary role in the choice). 

Remark: <a href="https://www.youtube.com/watch?v=VUdFdlQNfpg">Here</a> is Wildberger's reading of his paper on Youtube, for those who have no subscription for Intelligencer.