While I respect your point about rigour I should say one has to be careful not least because the way the subject is presented in textbooks are most often backwards -- axioms are really an end not the starting point.

Here's another view: Euclidean geometry is euclidean because the underlying transformation group (the group of those transformations which preserve what we want to preserve -- in this case the metric) is the euclidean group (the semi-direct product of the orthogonal group and translations). This is the symmetry that encodes our intuition -- the same intuition the kid is using to prove the above result. If we were to change the underlying space to the real projective space instead of R^2, and instead of choosing to preserve the metric we choose incidence and cross-ratio, we'd get a different group (GL(3,R)) and different geometry, viz. projective geometry.

This is an ancient dialectic that runs within mathematics -- embodied in modern math by Hilbert on one side (the formalist) and Poincare on the other.