Shipboard discipline in the Imperial Charisian Navy
For the dialogue I should explain that it's a bit of a cumulative effect. Something I've noticed across many books, and even different series like the Bahzell books is that once somebody joins "Team Good Guy", when they are talking to anybody else on team good guy all distance, difference of opinion or perspective, or rank seem to disappear and everybody sounds like they are chums shmoozing at a cocktail party. (Sometimes you'll have a minor difference of opinion, so that the characters can discuss an infodump.)
For me, at least, it stands out like a clown at a state dinner. Characters may have different accents, or mannerisms of speech but basically they all seem to be clones with identical viewpoints, opinions and social standing.
On a warship, at sea, in an emergency it stuck me particularly because everything I've ever read on the subject (which is not just fiction, although I've read both the Hornblower and the Aubrey books) says that the 'Master after God' of a ship at sea is probably the loneliest man on earth. Because even if he likes and loves the crewmen and they like and love him back he is not, and cannot be friends with any of them. That's not his job, nor his place. That's why Maturin is Aubrey's only friend on a ship, because the doctor is not a crewman and is not subject to orders. And most captains don't even have the consolation of that one friend out of a crew of dozens or hundreds with whom he lives cheek by jowl.
This might be less true of a merchant ship perhaps, and often untrue of a pirate ship which were (for a while there) practically the only democracies on earth, with the Captain merely being a job to command in battle. But a warship? Granted Safehold is not earth, and the Charisian Navy is not the British Navy but shipboard discipline was there for some damned good reasons and from what I understand the brits were actually almost easy going by the standards of the day, as ridiculous as that sounds to the modern ear.
Shipboard discipline is there for a reason, indeed. And the captain in this instance is speaking to his two most senior warrant officers. This is like the captain of a nuclear submarine talking to the chief of the boat, and commissioned officers --- yes, even COs --- speak to the command sergeant major in rather a different way than they would speak to, say, a recruit seaman or a Pfc.
As for your general observation about the captain being the loneliest man in the world, this is both true and not true. It has been seized upon in a great deal of naval fiction because it emphasizes the massive burden carried by the commanding officer of a detached vessel thousand of miles and months of sailing time from any friendly port. In truth, every commanding officer commands in the style which he believes works best for him, and captains like Nelson and Alexander Cochrane --- both of whom truly were notoriously "lax" where harsh discipline was concerned, both of whom were on a first-name basis (from their side) with many of the enlisted men under their command, and both of whom were deeply beloved by the men who served under them --- were quite social animals where their officers and senior warrant officers and petty officers were concerned.
The degree of harsh discipline practiced in the Royal British Navy during the Napoleonic Wars (and it was most assuredly not "almost lax" by the standards of the day) was not the only --- or even the most effective --- form of discipline in the world. The USN ships which won such a remarkable succession of victories against the RN in the early period of the war of 1812 were "lax" by British standards, and the Danish Navy was even worse (by British standards), but the quality of their basic manpower was also much higher. To man the ships of the RN, the press gangs swept up anyone they could catch, and a degree of discipline which could only be called brutal was often necessary (or thought to be, at any rate) to hammer them into the crews who won the RN's battles and wars. (It was also thought necessary to provide them with enough rum that many of them spent their time in a continuous alcoholic haze which was a major contributor to deaths from such things as falls from aloft, but I digress.) US seamen in 1812 were all volunteers, which was also the case for the Danes, and that led to a very different situation aboard ship.
To some extent, the ICN is in the same position as the British Navy, but it is not the British Navy and, up until the end of the previous book, has managed to crew its ships on a volunteer basis. That may well change in the very near future, but it hasn't changed yet, and Yairley's crew has been serving under him without interruption for years now. The situation aboard Destiny is going to be akin to that aboard a US submarine on a deployment in 1943-44, except for the fact that a Gato had a complement of around 60 (IIRC) and Destiny has a crew of 350-400. Perhaps a better example would be a Fletcher class destroyer like the Johnston at Leyte Gulf, since her ship's company was around 300. You might look up Commander Ernest Evans, Johnston's CO sometime and take a look at the command style of an officer who led his men through a battle against, frankly, unbelievable odds, and never seems to have had a bit of trouble maintaining his authority, whether under more peaceful conditions or when his ship was literally sinking beneath him under fire from an entire squadron of Japanese destroyers at pointblank range.
As far as the attitudes of my characters standing out like clowns at state dinners, you obviously have a right to call it however you see it. If you'd prefer to read someone else, you have a right to do that, as well. I will only add this: I have been told by the following that I get it right ---
(1) skipper of a Los Angeles attack submarine;
(2) skipper of a Montana-class SBN;
(3) skipper of an amphibious assault ship;
(4) engineering officer of an SSN;
(5) at least a dozen senior NCOs, Navy, Marines, and Army (haven't actually discussed it with an Air Force E6 or above to the best of my knowledge);
(6) a retired vice admiral;
(7) a holder of the Navy Cross;
(8) a retired gunnery mate who served 3 war patrols on a Gato during WW II;
(9) Pilot of a carrier-based F18; and
(10) an Army special forces officer who served continuously from 1955 (when he was drafted) until 1993, when he retired with the rank of colonel.
Now, obviously they could all be wrong and you could be right, but I kind of think I'll put more faith in their opinion of how command relationships work out in the field.
Edit: to correct "1912" to "1812." Stephen Decatur would have been a little long in the tooth by WW I, I suspect.