I am wondering if the fact that "the electric potential is continuous" should not be added to the set of axioms of physics.
I have never seen a problem where the potential is discontinuous, unless, possibly if the problem involved non physical assumptions (like infinite objects etc.). Are you aware of any real situation where the electric potential can be shown to be discontinous?
There is no "the set of axioms of physics". Assumptions in physics are often adopted or rejected flexibly, depending on their utility for the problem at hand.
An example of a discontinuous electric potential is electric potential due a point dipole, at the point where the point dipole is. Depending on the direction from which we approach the dipole, potential can remain always zero, or grow negative, or grow positive, and thus at the dipole, it has no universal limit, and thus potential is not continuous there.
But this "problem" is only at that single point, everywhere else it is continuous.
Point dipole is so useful a concept that the discontinuity did not prevent people to adopt it into EM theory.
It is discontinuous if there ever exists a sufficiently singular charge density, which is extremely common. For example, point-charges with charge density $\sim \delta^{(3)}(\vec x - \vec x_0)$.
You might argue that these singular charge-distributions are unphysical, and merely crude approximations to smooth underlying charge-distributions. The key is to realize that this doesn't really matter - Maxwell's equations work just fine with such singular charge distributions.
Physics is based on phenomena, not axioms. Some mathematical models of physics have axioms, though. You are free to add your axiom to electromagnetic models since there is no experiment that can falsify it. I think most of us would rather not do so, as it would simply prohibit useful idealizations without adding any utility.
