We have a running Graphhopper Server in our company for getting routes.
Is there a way to find out how far we are driving in each country? e.g if I'm driving from Munich to Vienna I would like to know how many km we are driving in Germany and how many in Austria.
This is not yet possible out of the box. You'll have to create a feature request for this or implement this on your own: grab the country boundaries, identify the edges/nodes to mark in the graph and then create additional instructions or store the point indices to later calculate the distances/times for every path.
We now have built our own solution which displays us the distance per country and also separates which road types are used.