We can consider a directed multigraph. A simple directed graph has the following criterion: A directed graph is Eulerian iff it is strongly connected and every vertex has equal in degree and out degree. The same is probably true for a multigraph.

In our case, the criterion is true, so I believe the answer to the original question is yes.

A remark: To maximize chances for a helpful reply, it may be a good idea to explain some notions that are not central to the problem. In this case, one may know about Eulerian graphs but not about Königsberg.