Disclaimer: original posting made wrong statements about geometry, but I opted to preserve and fix it because a different approach ultimately resulted in similar conclusions that of other answers.
Addressing the second part (Euclidean) of the question.
There is very obvious parametrization of (Euclidean) circles: each has the centre (a point on the plane) and radius (a number), so as a smooth manifold their space is E2 × (0, +∞) (if we exclude all degenerate cases). Let’s use respective real coordinates $(x,y,r)$. To address the question of its geometry one first should determine which symmetry group act on it.
Which transformations of Euclidean plane preserve all circles? Obviously, similarity transformations. The next interesting question is a nature of their action, and the following observation will help us:
Any similarity transformation preserves quadratic forms (or symmetric 0,2-tensor fields) $\frac{1}{r^2}(dx^2 + dy^2)$ and $\frac{1}{r^2}dr^2$ on the space of circles.
From these forms a pseudo-Riemannian structure can be constructed.