According to the showrunner, Mike McMahan, the show is entirely canon within the Star Trek universe. This has been stated numerous times across multiple interviews without correction by the studio.
LOWER DECKS TAKES PLACE DURING THE NEXT GENERATION ERA. HOW DOES IT CONNECT TO CAPTAIN PICARD AND THE REST OF THE STAR TREK UNIVERSE?
Lower Decks is very different from Solar Opposites and Rick and Morty. Within Lower Decks, there is a proper in-canon Star Trek show. It takes place during the TNG era. It's on a ship that feels like it’s always existed there and the bridge crew is dealing with big, never before seen Starfleet Star Trek type stories. So every episode has a thing like that happening in it. And then, on top of that, we've got A stories and B stories that are emotionally driven from the point of view of the lower deckers on the ships. So it was an area of storytelling that people had covered every once in a while on Star Trek, but never built a show around.
It was important to me that if you know everything about Star Trek and you watch this show then it fits into Canon and doesn't break Star Trek. In fact, it grows it. And if you know nothing about Star Trek, then all of the canon in Lower Decks feels like mythological, broad understandable sci-fi stuff. So you can still enjoy Lower Decks even if it's your first Star Trek show.
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“It's important to me that canon and Star Trek really go hand-in-hand,” McMahan explains. “It's important, to me. It's not worth making a Star Trek show unless you are at least trying to make sure that it fits into the canon because the canon is part of why I and everybody else loves Star Trek. It feels like, that is what is kind of the original shared world of all these different shows. Now you're seeing it in the Marvel movies and the DC movies and all of that, but in the very beginning you had [Star Trek: The Original Series] and then it became TNG and all the other series reference each other, and the movies reference each other.
“To me, that was the original fandom and if you're not trying to fit into that stuff and not being careful, it's kind of not worth doing. That's part of the joy of doing it. It's not a constraint. So the trick with Lower Decks is that our characters are, our stories and our characters are definitely in canon. Ours, they're just a little bit more aware and self-referential. Our guys might talk about the characters from the other shows that they've heard about because they are as big a fan of Star Trek as I and my writers are. They don't get to do as important stuff. It's almost like our characters are aware that they're in canon, and some of them are geeking out over it.”
Star Trek: Lower Decks Creator Reveals How the Series Fits Into Canon
We also have confirmation (via a panel at Comic-Con@Home 2020) that the show is taking place in the Prime timeline, directly after the events of Nemesis
"You know, what you're about to see is people talking about a show that takes place in 2380, it's in the TNG era, it's right after the events of Star Trek: Nemesis, but it's way before the events of Star Trek: Picard."
As to whether everything should be taken hyper-literally, the answer is an obvious no. The show is a comedy and that includes the occasional easter-egg and nugget of self-reference.
For me, Lower Decks doesn’t betray Star Trek. We bend Star Trek rules. We are obviously a comedy, but at its heart, this show is Star Trek; everything from the design choices to the music to the words that the characters are saying.
Mike McMahan Explains Why ‘Lower Decks’ Has So Many Star Trek References; Plans To Go Beyond TNG In S2