Okay, so I'm not gonna defend this because I mostly agree that this is petty complaint of dumb exposition but the key difference is this is Obiwan, whose deal we know because at the time he was only in one other movie which virtually all the audience has already seen, vs Bo Katan, who mainly the hardcore fans know. For example, I haven't seen clone wars, so I didn't know her deal of what being a deathwatch means or how I'm supposed to feel about her. As far as the Mandalorian informed me, she's a mandalorian like Mando, but slightly different and she just gave him this info as a reward for the sidequest that they were just on, which has no real implications for Mando besides getting that info.
There's no getting around the fact that Mandalorian is really leaning heavily on the part of the audience that watched Clone Wars and Rebels, while ESB only leaned on the audience that saw A New Hope, which was, again, basically everyone who was watching ESB.
Also, it's worth noting the placement of these two lines in each respective narrative. Obiwan says this at the start of the movie. Mandalorian drops this line at the end of the fourth episode of the second season to close it out and will only answer it in the next episode after that. ESB uses it as a jumping point to propel the story of the movie, Mando uses it to conclude it's episodic story of a show.
Look, of all the shit Mandolorian does that isn't impressing me, this is extremely minor and people are making too big a deal out of it. If nothing else, if you're watching a Star Wars spin off TV show in the year of our Yoda 2020, and you're a season and half through it, maybe it's not that unreasonable for the show to spam you with some Star Wars jargon, even if it is a bit awkward when you step back and listen to how it in a normal, out-of-star-wars context and think of what your life is and how disappointed your parents are in you.