Formal but "Conversational"
To read The Remains of The Day is to be trapped inside Stevens's weird little head. And Stevens, with all his talk of perma-dignity and staying professional 24/7 (exhausting), talks to us in the manner in which he'd talk a coworker. He keeps his formality game strong.
Sometimes Stevens will make reference to "you," like he does in the first chapter:
"If you are not familiar with Mrs. Jane Symons's books—a series running to seven volumes, each one concentrating on one region of the British Isles—I heartily recommend them." (1.18)
TBH, Stevens is a bit of a robot. The quote above is almost creakily formal ("I heartily recommend them," "running to seven volumes"), but nevertheless is Stevens's concerted effort to try to be chummy. He's trying, guys. He really is trying. He's just not succeeding very well. He's suggesting a book to us with the same level of formality that he would use if suggesting his employer try on a new dinner jacket, or some rare claret, or some pheasant under glass (or whatever else English aristocrats in the 1930s liked).
Oh, Stevens. You're an awesome character, but jeez would we hate to be trapped in conversation with you on a cross-country flight.