Let's start with a concrete example. Can I assume that for a metric $d(x,y) = \sqrt{|x-y|}$ on $\mathbb{R}^n$, there is only one possible norm to be assumed to induce this metric, say $\sqrt{|x|_2}= \sum_k x_k^2$? I know that this is not a norm, but how can I possibly know, there is no other possible candidate?
More generally, what can be assumed about the inducing norm for a given metric? I'm guessing it might even boil down to isometry.
By definition a norm $\|.\|$ is induced by a metric $d$ if $\|x-y\|=d(x,y)$. Putting $y=0$ we get $d(x,0)=\|x\|$. So there cannot be any other norm than $d(x,0)$ which induces $d$.
If your question is : under what conditions on a metric $d$ does there exist a norm inducing it then the answer is simple:
a) $d(cx,0)=|c|d(x,0)$
b) $d(x+y,0) \leq d(x.0)+d(y,0)$
These two conditions are necessary and suffcient.