I didn't spot that. Yes, it should. But because it is in a different way it can't just be dragged. https://help.openstreetmap.org/questions/66522/how-do-i-copy-the-tags-from-a-node-to-a-relation-in-josm tries to explain how to do it, but I haven't managed it. So I'll have to leave that to somebody more familiar with relations.