It is related to commutativity: if there's an identity element, then it implies it (by putting $z=e$), if we have associativity, then it is implied by it (obviously). On the other hand, the property along with commutativity implies associativity (it is easy to check), so a loop with this property is already a commutative group, and any magma with identity and this property is automatically a commutative monoid.
Any finite set with operation of right projection $a\cdot b=b$ has the property and is associative, but not divisible and does not have identity (if the set has at least 2 elements), so that takes care of two arrows. (It is also not commutative.)
A finite linear order with the operation $a\vee b=\max(a,b)$ is a noninvertible monoid with the property, which takes care of the lower right one.
As per Generic Human's answer, a simple example of a quasigroup without identity (albeit with a left identity) that is nonassociative, but having the property in question, is the cyclic group $\mathbf Z_3$ with the operation $x\cdot y=y-x$ (or any commutative group with an element of order at least 3 with the same operation), so that is the complete picture.
I'm opening a community wiki, so others can contribute.