I have seen this partial order before (for a specific context) in Hart J.E., Kunen K., *Bohr compactifications of discrete structures*, Fund. Math. 160 (1999). Let me briefly explain the situation considered there:

Fix a set $A$. A pair $(X,\varphi)$ is a *compactification* of $A$ if $X$ is a compact Hausdorff space and $\varphi:A \to X$ is a (not necessarily injective) function such that $\varphi(A)$ is dense in $X$. Define $(X,\varphi) \leq (Y,\psi)$ if there exists a continuous $f:Y \to X$ such that $f \circ \psi=\varphi$. Note that the last condition implies that $f$ is surjective since $f(\psi(A))$ is dense in $X$.

Now just as you did, they define the equivalence relation and the partial order on the set $\mathbb{K}(A)$ of all equivalence classes: $(X,\varphi) \sim (Y,\psi)$ if and only if $(X,\varphi) \leq (Y,\psi)$ and $ (Y,\psi) \leq (X,\varphi)$; $[(X,\varphi)] \leq [(Y,\psi)]$ if and only if $(X,\varphi) \leq (Y,\psi)$. The following is Lemma 2.2.7:

$\mathbb{K}(A)$ is a complete lattice.

You can also add a topology and/or a first order structure to the set $A$ and study a restricted version of compactifications. For instance this gives a way of defining the Bohr compactification of a topological group.