I think it should be a tree since it is acyclic and connected, but the crossing edges $B - F$ and $C - E$ make me unsure about it - it doesn't look like a "classical" tree, intuitively (presumably because by "classical" tree, I have directed and rooted trees in mind, although I am aware that undirected trees exist, too), and I can not recall having seen such a graph referred to as 'tree' ever before.
Which graph-theoretical classification would apply to this object? Are the distinctions directed vs. undirected, and ordered vs. unordered of relevance here?
Edit: I am thinking that ordering might be of relevance because if you were to "untangle" the lines and align the nodes in such a way that node $F$ is right next to $D$ and $E$ is right next to $G$, the linear order of nodes would be a different one, and that this would make it different tree. Is this correct?