This is a basic content fallacy where one is using human psychology instead of logic. It is akin to an ad-hominem fallacy, where the quality of a person as a human being is taken to bear rationally on the quality of his arguments. And in quality, it lies alongside observations like "He keeps arguing with you because he likes you." (and not because he believes what he is saying is true.) The problem is that as reliable as some emotional reactions might be, they are logically off-topic.
In this case, Z is imagining X's logic is actually defensiveness, due to some quality of its presentation, perhaps its length or tone, and one only defends oneself vigorously against arguments one considers likely to have an element of truth. So he presumes Y's argument is a valid threat to X, and therefore that at least part of it must be true.
But X may be legitimately scared or angered by some element of Y's argument instead of defensively so. Or X's own opinion of the arguments' relative strength may be flawed. Or X may simply be coloring his arguments emotionally for rhetorical effect, or for other reasons entirely. Whatever the reason for the observed emotion, if it is even present, and not Z's imagining, it is not logically connected to the validity of X's argument.