Is there a term for a variable that is both a mediator and a confounder at the same time? By this a mean a variable that both influences the exposure and the outcome but there is also an influence from the exposure to the variable as indicated by the diagram below:
Your example is not a Directed Acyclic Graph (DAG) because it has a cycle in its bi-directed edge. So in terms of causal graphical models, I do not think there is a name for such a structure because it contradicts the underlying causal DAG assumption. That said, it might make sense to consider common causes of "Exposure" and "Variable", or find other ways of expressing your set-up in a graph that does not rely on a bi-directed edge.
If you really do believe there is a bidirectional effect that cannot be resolved either through a different model, or by arguing things like "the effect in one direction will take very long to show so it does not matter", then it seems the DAG modeling assumptions are simply violated. I should say that this would not be unusual - they are very strong assumptions. There is also some theory on cyclical causal graphs. I am not sure how applicable such approaches would be to your case, but it could be an idea to look into it.