What kind of order is binary tree ancestry?
Your condition is almost equivalent to the statement that the partial order is a tree.

Consider the following two conditions on a partial order $\ge$:

C1: There exists a top element $r$ (also called "the root") such that $\forall x . r \ge x$.

C2: For all $x,y,z$, $x \ge z \land y \ge z$ implies $x \ge y \lor y \ge x$ (this is your condition).

Notice that every tree satisfies both conditions (i.e., its isAncestor partial order satisfies both of these conditions).

Moreover, I claim that every partial order that satisfies both of these conditions has a Hasse diagram that is a tree.

Why?  Well, if C1 is true, then C2 is equivalent to

C3: For all $v$, there is a single unique path from $v$ to the root $r$ in the Hasse diagram of $\ge$.

This is easy to prove by looking at the contrapositive: if there were two incomparable paths from $v$ to the root, then we could find a violation of C2 (let $x,y$ be two incomparable elements, each on one of the paths).

And if C1 and C3 is true, then the Hasse diagram is a tree.



Bottom line: there's no particular reason to expect any special name for your property.  Your property almost characterizes trees, once we take into account the correspondence between a partial order and its Hasse diagram (which is a DAG).