Not necessarily, and not all of it. X-Men continuity is tricky.
In general, it's hard to reconcile the X-Men Cinematic Universe together no matter which movies you use. Origins: Wolverine contradicts a lot of things, Days of Future Past reboots things, etc. In this case, there were vague and contradictory statements abounding about Logan, some implying that it was its own universe, others that it followed directly from Days of Future Past's end scene, etc.
One thing we do know is that Fox does not consider X-Men Origins: Wolverine to be canon anymore. Fox executives went on record as saying that, part of the motivation for Days of Future Past was to try and "clean up" some of the continuity mess that Origins had left. So we can ignore the fact that an adamantium bullet only injured Logan but killed X-24.
How to reconcile The Wolverine and Logan isn't as clear, but again, the situations aren't exactly the same. The Samurai was trying to slice through Logans claws with his own blades; a bullet fired at close range has much more power behind it, and is already heated from the gunpowder.
We also don't know that X-24 has exactly the same adamantium structure as Logan; perhaps Alkali decided to leave a few Achilles Heels in him just in case he ran amok the way Logan did.
As a whole, there may be parts of prior movies that we have to erase or explain away as non-canon anymore, but the general timeline of events and the characters that took part in them is still mostly intact.