What is the name for this relation between metric spaces? Consider two metric spaces on the same set $(X,d_1)$ and $(X,d_2)$ such that for all $x,y,z$, we have
$$ d_1(x,y)\leq d_1(x,z) \Leftrightarrow d_2(x,y)\leq d_2(x,z) $$
Is there a certain name to describe when two metric space has this relation?
 A: It is convenient to formalize relations between metric spaces in terms of maps between them. When both spaces are defined on the same set, the map can be taken to be the identity. For example, two metrics $d_1$, $d_2$ are comparable ($C^{-1}d_2\le d_1\le Cd_2$) if and only if the identity map $f:(X,d_1)\to (X,d_2)$ is bi-Lipschitz. 
In your case, the relation is that the identity map $f:(X,d_1)\to (X,d_2)$ has the property 
$$ d_1(x,y)\leq d_1(x,z) \implies d_2(f(x),f(y))\leq d_2(f(x),f(z)) \tag1 $$ 
(You wrote the implication as $\Leftrightarrow$, but one direction gives the other by exchanging $y$ and $z$.) 
Maps that satisfy (1) are called $1$-weakly-quasisymmetric. More generally, $f$ is $H$-weakly quasisymmetric (where $H$ is some constant) if the following holds: 
$$ d_1(x,y)\leq d_1(x,z) \implies d_2(f(x),f(y))\leq H d_2(f(x),f(z)) \tag2 $$ 
In connected doubling metric spaces weakly quasisymmetric is equivalent to quasisymmetric. This result was proved by Väisälä and can be found in section 3 of  QC maps in metric spaces by Heinonen and Koskela.  
Weakly quasisymmetric maps were not studied much on their own (except when they are in fact   quasisymmetric). Notable exception is the paper "Bounded turning circles are weak-quasicircles" by Daniel Meyer: it deals with metric spaces that are not doubling, where the distinction between quasisymmetry and weak quasisymmetry becomes real.  

In another direction, (1) is a restricted form of the 4-point inequality
$$ d_1(x,y)\leq d_1(w,z) \implies d_2(f(x),f(y))\leq d_2(f(w),f(z)) \tag3 $$
Maps satisfying (3) were called monotone by Bilu and Linial. This is not the most common usage of the term monotone map:  it means another   thing in topology, and yet another one in functional analysis. But if you were to stick with it, you could call condition (1)  3-point monotonicity, because it amounts to monotonicity on every 3-point subset. 

Short version: "no, there is not a name that you can expect people to recognize".  
