The property $(xy)x = xy$ is one of the equations satisified by a directoid. Various properties have names ($xy = yx$ is commutativity, $xx=x$ is idempotency, etc). The wikipedia page for Magma has a long list of named properties, but not this one.
Directoids also satisfy $y(xy)=xy$ which is similar to the one above (although which should be called a 'left' property and which 'right', I am unsure). Last, there is $x((xy)z)=(xy)z$ - for which I have also not been able to find a name. Have these been named?