Let $(X,d)$ be a compact connected metric space with the property that for any distinct points $a,b$, $X\backslash \lbrace a,b\rbrace$ is disconnected. Clearly the unit circle has this property. Is there any other example (up to isomorphism) ??
This property indeed characterizes the circle, but this is not obvious.
This was shown by R. L. Moore, according to Sam Nadler's Continuum Theory p. 156.
Added: the precise reference is  in this historical survey of continuum theory.
Yes and this is a nontrivial result, see e.g. J. G. Hocking, G. S. Young, "Topology", page 55, theorem 2-28. See also their theorem 2-27 for topological characterization of the interval.