How to get current name of route in Vue?

You are using computed incorrectly. You should return the property in the function. See the docs for more information.

Here is your adapted example:

computed: {
    currentRouteName() {
        return this.$route.name;
    }
}

You can then use it like this:

<div>{{ currentRouteName }}</div>

You can also use it directly in the template without using a computed property, like this:

<div>{{ $route.name }}</div>