Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$.
Say that a metric space $X$ has the *superposition property* (my terminology) if, for every pair of isometric subsets $A$, $B$, there is an isometry of $X$, $F: X \to X$, that superimposes $A$ onto $B$: $F(A) = B$, i.e. $F$ places $A$ on top of $B$.

Which metric spaces have this superposition property?

Euclidean space $\mathbb{E}^d$ has this property. But it seems the punctured plane does not: e.g. if $A$ is the point $(1,0)$ and $B$ is the point $(-2,0)$, then (I believe) there is not an isometry of the whole punctured plane that maps $A$ onto $B$.

Has this property been studied before? If so, under what name? I am (clearly) unschooled in this area. Thanks for pointers and/or examples!