(needs some fixing)
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 the metric $\frac{1}{r^2}(dx^2 + dy^2 + dr^2)$ on the space of circles.
Of course, we identify this metric as Poincaré half-space model of hyperbolic 3-space, but are all isometries of hyperbolic 3-space reasonable transformations of Euclidean circles? 3-spaces have 6-dimensional motion groups, whereas similarity transformations amount only 4 dimensions. What are two “lost” dimensions? Let’s simplify our quest by considering the $y=0$ plane in the circles’ space and will find one lost dimension in its motions.
“Correct” transformations of our $(x,r)$ subspace are generated by two 1-parametric subgroups: $$(x,r)\mapsto(tx,tr)\quad(x,r)\mapsto(x+c,r)$$ homotheties translations
A subgroup generating the rest of (orientation-preserving) motions of Lobachevsky’s plane would be, for example, rotations about certain (non-ideal) point $(x_0,r_0)$. I will not bother to write these transformations explicitly, just a qualitative observation: such rotation can transform a circle (nearby $(x_0,r_0)$) from lying strictly inside to one intersecting with $(x_0,r_0)$. Hence, (most of) these transformations do not respect what we expect of these $(x,r)$: to be (a parametrization of) circles.
Therefore, the answer is:
The space of circles on the Euclidean plane is a hyperbolic 3-space with certain restriction (possibly will be clarified later) on its motions.