When is the minimum spanning tree for a graph not unique
in the first picture: the right graph has a unique MST, by taking edges $(F,H)$ and $(F,G)$ with total weight of 2.

Given a graph $G=(V,E)$ and let $M=(V,F)$ be a minimum spanning tree (MST) in $G$. 

If there exists an edge $e=\{v,w\}\in E \setminus F$ with weight $w(e)=m$ such that adding $e$ to our MST yields a cycle $C$, and let $m$ also be the lowest edge-weight from $F\cap C$, then we can create a second MST by swapping an edge from $F\cap C$ with edge-weight $m$ with $e$. Thus we do not have uniqueness.