I was wondering if there is a way to highlight a feature (e.g. line) only by selecting its label on the map.
Can ArcMap autmatically highlight this object or do I need to create some "connection" between line and its label?
Not with labels. You can sort of do this with Feature Linked annotation. This creates a relationship between the annotation and the related feature. During an edit session, by selecting a piece of annotation you can see the related feature in the Attributes panel. You can then right click the related feature in the Attribute feature and select it.
Other than that, this would require a custom tool.