Currently, Aurora randomly selects a theme for an NPR, then randomly selects a(n associated) theme for a class, then procedurally selects a ship name from that class.

Then, when you encounter said NPR, Aurora again randomly selects a theme by which to report it to you, randomly selecting a(n associated) theme for reporting the class, and appending an ordinal number for the unit.  It also gives you an option to just use the names the NPR is using.

So your complaint is that the randomly selected name & class that the (usually alien) NPR is using makes no sense by your empire's rules, but because it's called the 'real name' you feel unable to ignore it.

I'm pretty sure the Tac Intel window allows you to rename your reporting names for other empires/ships/classes (though I'm not going to fire up Aurora just to check).  I advise you to turn off 'use real names for alien ships (if known)' and pretend that option doesn't exist.

If what your asking for is a way to 'push' player-designated names onto NPR ship/class designs, I would advise against it because that seems to me to be begging for an endless supply of bugs from accidentally including an apostrophe or ampersand or other special character in a field that does not get properly sterilized.

Sloanjh has the right idea.  Even better, I think his "Change reporting name" button already exists, albeit by a different name, on the Tac Intel and/or Diplomacy windows

I would be happy if it randomly picked the name in the class, not just alphabetical order. I am sure that is something easy to code.

Sure, that would be a good compromise indeed. Have me able to change how a class of ships (owned by NPR) get its names from would be a major advance already. Instead of manually renaming them after I figure out their class.
But that would provide me an instant extra info? If I decide all AI Battlecruisers should pick from the list of German pocket battleships... But then, is this a problem?

I think Carson C1 #017 is perfectly fine. I don't want longer civilian ship names on my screen and they would anyway cycle every couple of years as ships are replaced.

Could be a display toggle, and I do think the idea has merit since one key aspect in Aurora is to tell stories, and individual ship names certainly helps with that.

Bear in mind that the US Navy have a transport ship named Bismarck... 

Saying that, I would like it if each civilian line picked a name list at random and used it (with the companybname) for their  1st. So instead of Carson C1 #017, you might have the Carson Wyoming (C1).

Except ... except if I check the box 'use real names' in Tac Intel window!

To be precise, the problem is not with me, the player. I can definitively name correctly my ships 

The problem and issue is with the NPRs. They don't have a clue that 'Bismarck' should not be a tug but an heavy combatant (at list CA or heavier). I can understand it is tricky to code rules that are not too complex (code-wise), intelligible (for the user) while being smart enough to do the job. But I felt I had to raise the point, and Steve will decide whatever he likes, sure.

Go try that Class design->DAC/Rank/Info->Name type for automatic naming of new ships.