It's mostly due to them being commoners and the king is of a higher station then them. So while Yuna treats him as just a person, ordinary people of the country will have a harder time to relax around them. Especially when there are other nobility that like to throw their weight around. So while the king is kind and benevolent if he hasn't been interacted with by commoners then the bakers reaction is quite normal
He strikes me as the kind of king that would be well known to the people as a decent guy, I'm not saying he and the peasants get together and sing kumbaya and roast marshmallows or anything too formal but he punishes nobles that overstep and even snuck out for pudding like a teenager.
All in all I just feel that their reaction is a bit too fearful and not just overly formal, I expect them to bow in his presence, not cower in fear like he is a brigand, they were more afraid of him than the actual thugs that wanted the daughter for a fun night (for them).
TLDR: I agree that they should not be able to relax around the king but think the fear was too exaggerated.