I'm re-reading Fugitive Telemetry and have some thoughts. On Preservation, when Murderbot meets free bots, it's disdainful of them having names, especially if they're "infantilizing". It practically has a seizure when it meets Preservation's silliest cargo hauler, JollyBaby.
When asked by officer Indah to set a feed ID, it at first refuses, but eventually sets it to SecUnit as a tongue-in-cheek way of giving her what she really wants: for it to disclose what it is, so people will be aware of the "danger" it poses. Murderbot, I think, doesn't actually intend for this to be recognized as a name, but it's what it's been called all its life by clients and it at least means that it doesn't have to pretend to be an augmented human, which it admits is difficult for it.
It never questions the fact that any humans have names- this is simply fact. So why does it find it so difficult to think about itself or other bots sharing their names? I think it's about not wanting to be a pet bot- not stooping to identify yourself in a way that's simple enough for humans to use, unlike a hard feed address. It sees it as demeaning that it or any bot should have to change the way they communicate to suit humans.
Why, then did it bother to choose a name for itself? Beyond that, why does it bother to choose names for other bots it interacts with, notably ART, Transport, and Ship? It sees it as embarrassing that ART's humans have a "cute pet name" for it, even though it refers to it by its own pet name, when surely it was listed in the transit ring as the Perihelion, or it could use ART's hard feed address, as we know ART does for Murderbot. This is definitely some form of endearment, because it never names bots, SecUnits, or CombatUnits that it's fighting. They're always just Hostile or Target, which are less names than categories. I think it's in some way part of the cognitive dissonance that comes with being a construct: simultaneously resenting, even looking down on humans, but finding it natural to communicate in the same ways as them. It does want that human aspect of connection (even if it would never admit that), so it bestows a little humanity on its bot allies in the form of a nickname.
I think its rationale for choosing its own name is much the same. When it freed itself, it probably wanted some secret power to itself, to prove to itself, in defiance of the company, that it was a person in the way that watching Sanctuary Moon made it feel like a person. It probably also couldn't imagine itself with some boring human name that would only remind it of its captors, and remind it of the shameful human part of itself. Instead it chose one of the two "construct genders" (murderbot and sexbot), which is probably the only other term that had ever been used to refer to it directly, aside from SecUnit. It's self-deprecating and a little bit of an inside joke with itself, to downplay the importance its name carries and the complicated emotions it has around its relationship with humanity. That's probably why it considers its name private- because it's a little embarrassed to have given itself one in the first place. It's an admission of a desire to connect with people, and at the same time a warning of all the reasons it shouldn't.
