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The situation is impossible if we make the following assumptions:

Each circle has exactly one center, which is a point. Concentric circles have the same center. Each circle has exactly one radius, which is a number. (We make no assumptions, besides those listed, about the meaning of the word "number.") If two circles intersect each other at a point, then that point lies on both circles. Given any unordered pair of points, there is exactly one distance between those points, which is a number. If a point $p$ lies on a circle, then the distance between $p$ and the center of the circle is the radius of the circle.

From the above, suppose that $C$ and $D$ are two concentric circles that intersect at a point. The two circles have the same center, $e$, and call the intersection point $p$. Then the distance between $e$ and $p$ is the radius of $C$, but it is also the radius of $D$, so the two circles cannot have different radii.

We could make the situation possible by discarding some of the axioms, but for the most part, these axioms are so fundamental to the notion of geometry that if you discarded one, the result wouldn't be considered geometry any more (not even non-Euclidean geometry). In particular, axioms 2 and 4 above are essentially just the definitions of the words "concentric" and "intersect," and axioms 1, 3 and 6 essentially constitute the definition of a circle with a given center and radius.

If I had to pick an axiom to discard, I would discard axiom number 5: the statement that given two points, there is only one distance between those points. This is the approach taken in Luca Bressan's answer (a plane based on modular arithmetic, where pairs of points with a distance of $0$ also have a distance of $2$ and vice versa) and in Yly's answer (a cylinder, where a pair of points has infinitely many distances, depending on which direction and how many times you wrap around the cylinder as you measure the distance).