One possible answer that I just found thanks to a GitHub issue is the following.
It is possible to use the key attribute that is also used for v-for to let Vue track changes in the view. For that to work, you have to add the attribute to the router-view element:
<router-view :key="$route.fullPath"></router-view>
After you add this to the view, you do not need to watch the $route anymore. Instead, Vue.js will create a completely new instance of the component and also call the created callback.
However, this is an all-or-nothing solution. It seems to work well on the small application that I am currently developing. But it might have effects on performance in another application. If you really want to disable the reuse of the view for some routes only, you can have a look at setting the key‘s value based on the route. But I don’t really like that approach.