Is there a name for relations with this property?

If the relation is seen as an undirected bipartite graph with edges joining $X$ and $Y$, the axiom is that a path of length 3 closes to a cycle of length 4.

In this form it is easy to check that "are both related to the same element in the other set (or equal)" is an equivalence relation on $X$ and on $Y$, and that if $x$ is related to $y$, all elements of $X$ equivalent to $x$ are related to all elements of $Y$ equivalent to $y$. This is the condition for there to be quotients of the sets $X$ and $Y$ on which the relation descends to a one-to-one correspondence.

This can be expressed by saying that $\rho$ is/induces an "identification between quotients" of $X$ and $Y$. That property is not closed under composition as far as I can see, but the other properties are true.

Such relations are called rectangular in Section 5.2, page 669 of Andrei A. Bulatov, Víctor Dalmau, Towards a dichotomy theorem for the counting constraint satisfaction problem, Information and Computation, Volume 205, Issue 5, May 2007, Pages 651-678. They in fact define the property for higher arities, but it simplifies to your definition for binary relations.