Let $G$ be an undirected graph. Define the 2-step graph $G^{(2)}$ of $G$ to be the weighted graph whose vertices are the same as those of $G$ but whose edges correspond to 2-step paths in $G$. Thus the weight of an edge $(u,v)$ is the number of distinct vertices $w$ such that $(u,w)$ and $(w,v)$ are both edges in $G$. (In particular, the weight of $(u,u)$ is the degree of $u$ for every vertex $u$.) My question:
Are there two non-isomorphic graphs $G$ and $H$ such that $G^{(2)}$ is isomorphic to $H^{(2)}$?
My intuition says that the answer should be "yes", but I'm unable to construct an example.
EDIT: This answer assumes a path cannot contain repeated edges or vertices. The OP is interested in the case where edges and vertices can be repeated. See comments below.
Here is an example of two non-isomorphic graphs $G$ and $H$ with isomorphic 2-step graphs $G^{(2)}$ and $H^{(2)}$.