$S = \{(x,y): x \text{ is a sibling of } y\}$. Why is this relation not transitive? It seems to me that when we define "sibling" to mean a brother or a sister, if person a is a sibling of person b and person b is a sibling of person c, then person a is a sibling of person c.

A sibling of B and B sibling of A, if transitive would give A sibling of A which is not the usual use of "sibling".