Starfleet just fought a war, so, it's either a military, or something close; I've always figured a hybrid agency - humanitarian and scientific in focus - but equipped to defend the Federation in the worst case.
I would have to watch the show again to provide comprehensive examples, which I'm not willing to do for the sake of a thread that will be forgotten in a couple of days.
If you can forgive me for picking lame examples (which will no doubt be taken out of context and used against me maliciously)... 1). take the line that Pike gave the Chancellor in this episode: "it's difficult to explain". How hard would it be for him to tell her that her son has grown up faster due to a time anomaly. Why is it hard? Emotionally hard? Then say "this might be difficult to hear, but your son has grown into an adult, thanks to the effects of the time crystals". 2). Why did Burnham conceal that she was seeing visions of the Red Angel from Pike, when this was highly pertinent information "Captain, I was injured, so may have been hallucinating, but I thought I saw an entity on the asteroid". That is all that needs saying. Really important duty to provide all possible pertinent information to your CO.
3). Why does Tilly never pause to create a cogent sentence? Is this an anxiety disorder, and if so, why wasn't she screened from Starfleet? Sadly not everyone is fit for service on a starship loaded with megatons of antimatter weapons. 4). Why did Burnham not inform anyone that Tyler was suffering major difficulties, in a very crucial situation operating a laser torch, in season one? 5). Check out the science officer in episode one, who presented a barely heard piece of exposition about the red flashes, and subsequently got killed displaying absurd levels of arrogance in a life-and-death situation. These are just a few examples of a frequent tendency toward ridiculous characterisation of professionals and incompetent communication. Why are they so infantile at times? The contrast with say SG1 is especially bad, where people with a dizzy spell returning from a mission, would immediately be sent to medical for a check-up, as you would expect in an unknown situation like space travel - procedure was followed. Where Carter's scientific analogies actually aided clarity instead of sounding hollow.
On a positive note, Stamets gave a "like a baloon" type analogy a couple of episodes ago that actually worked, rather than being out of place. I am guessing some of the staff understand that you need to deliver technical ideas with some conviction, and others don't. Most such analogies do nothing to express things clearly, and are just there because someone think it's a twee Star Trek trope.