Eugenio is the Italian courier to the Miller family. He orchestrates their visits and travelling arrangements and waits on them as is customary for a travelling servant. That's where the customariness in this relationship ends.
The Millers tend to defer to Eugenio on manners of taste and etiquette—nothing could be more gauche. Their inappropriate relationship with Eugenio, who becomes very bossy and insubordinate as a result, is one of the hallmarks of their inability to "get it right" in the eyes of the old-money characters.