Understanding a metric with prescribed range I have the following question in my book:

Show that any subset $A$ of the set of non negative reals with $0 \in A$, is the set of all distances between points of some metric spaces.

And the solution is given to be: 
For a given set $A$ of non negative reals with $0 \in A$ , we define a function $ d: A \times A \rightarrow \mathbb R$ by  $$ d(x,y) =
 \begin{cases}
\text{max}\{x,y\}, & \text{if}  x \neq y \\
0 , & \text {if} x=y
\end {cases}
$$
Then $d$ is clearly a Metric on $A$ and for any $x \in A$ , $d(0,x) = x$ ; also for any $x , y \in A $ , $d(x,y) = \text{max}(x,y)$ or $0$ so that $d(x,y) \in A$. Thus $(A,d)$ is the required metric space.
Now my question is: Is the metric taken in the solution fixed? Can we take any other metric e.g the usual metric or any other metric on $A$?
My next question is there any special significance of the 'max' metric used in the solution on any space? By special I mean like the usual metric is the usual distance between points in the co-ordinate plane, such as if this metric has any significance.
 A: 
Is the metric taken in the solution fixed?

If you mean "is it unique", the answer is no. For example, take any bijection $\varphi:A\to A$ and let $d_\varphi(x,y) = \max(\varphi(x), \varphi(y)) $ for $x\ne y$.  More generally, let $B$ be any set of the same cardinality as $A$ and define $d_\varphi$ on $B$ as above, using a bijection $\varphi:B\to A$. This is described as "pulling a metric back by $\varphi$".
Also, the underlying set need not have the same cardinality as $A$. Given any set $X$, we can define a metric on $A\times X$ by 
$d((a,x),(a',x'))=\max (a,a')$ unless $(a,x)=(a',x')$. In particular, we  can put such a metric on $A\times \mathbb{R}$ and then pull it back to $\mathbb{R}$ by some bijection $\mathbb{R}\mapsto A\times \mathbb{R}$.

any special significance of the 'max' metric 

It has a reasonable interpretation as the required fuel capacity of a plane that has to fly from $x$ to $y$ with a required stopover at $0$, where it is refueled. 
It's also an example of a ultrametric, meaning $d(x,y)\le \max(d(x,z), d(y,z))$. Absent any assumptions on the structure of $A$, we pretty much have to use an ultrametric: given that we must have a triangle with two side length $a,b\in A$, what should the third sidelength of the triangle be? We don't know if any other elements of $A$ (if they even exist) can fit the triangle inequality. Using the smaller of $a,b$ risks violating the triangle inequality, if it happens to be less than half of the larger. So the only safe choice is to have the third side be $\max(a,b)$. Hence, an ultrametric. Notice that $\max$ function comes up naturally here.
