How to make JOSM show specific objects

How should I configure JOSM so that to show only highways with their own references?

You have at least two default options.

  1. JOSM has a complete filter option, see Filter panel.
  2. Using Overpass Download (second tab in download dialog) allows you to download filtered data.

Additionally, you could use an own map paint style which only highlights you combinations and could show all information you’d like to.