It's unclear because Matt Nix said one thing early on and now he is saying something else, but ultimately it seems to be an alternate universe...
Independent (from the Hollywood Reporter) January 25, 2017
Burn Notice showrunner Matt Nix is penning the script, having
previously said of the show fitting into the wider X-Men universe: "If
you look at the movies, which take place from — they started in 2003
to now [the first film was actually 2000] — they don't all line up
“I'm not slavishly fitting them into a particular slot. But at the
same time, if you like the world of the movies, there are definite
nods to the movies. It exists in the same general universe.”
x men tv show fox film tie in spin off legion
Comicbook July 21, 2017
As Nix explains it, "One of the great favors that ‘Days of Future
Past’ did for all of us is establish many streams. Guess that one
answer is that this is one of those many streams.”
Nix went on to drop an even bigger reveal, regarding how the show will
treat it's alternate universe "stream" of The X-Men team, stating,
"“Right now one of the things that comes out in the show is the X-Men
are gone. Why are they gone? That’s a thing in the show. It’s not just
that they’re too expensive for television — though that may be
Donner was asked by a fan if Gifted and Legion could be part of one
shared universe, but also doused that hope: “I’m sorry to break your
heart. No, but thank you for wanting it.”
x men gifted tv series movies connections
The Gifted is an American television series created for Fox by Matt
Nix, based on Marvel Comics' X-Men properties. It is connected to the
X-Men film series, set in an alternate timeline where the X-Men have
disappeared. The show is produced by 20th Century Fox Television in
association with Marvel Television, with Nix serving as showrunner.
So as far as I can tell it does take place in an alternate universe (like Legion), but clearly it is borrowing on the Logan idea, but even more so from Days of Future Past, as it features Sentinels, Blink, and Polaris (it's confirmed that this character is a certain someone's daughter and the exploration of that is apart of the character's story) juxtaposing the introduction of the "what would be" half sibling, Quicksilver.
However, with all of these elements, including that Days of Future Past is the introductory point for the X-Men films to introduce time travel and alternate reality, and because of the changes in terminology on when this takes place (which I do think was in part to manage our expectation), I keep thinking that perhaps the disappearance of the X-Men in this universe will somehow touch one of the films and is an important mythology point.
Just to clarify about Legion:
Legion does take place in an alternate universe/timeline and is done in a way that makes the reality seem timeless by not laying down a date so that the reality can convey a variety of aesthetic, most noticeably the "golden age" era, but with heavy allusions to British 60's and 70's [science-fiction] films. It's unclear if it is in fact timeless, but Noah Hawley did state that part of the how reality looks is really through the perspective of David more so then the reality itself. It is in a timeline where "X-Men" have never existed, where in The Gifted, they did exist, but they mysteriously disappeared.