The trailers for Morbius played it coy with where the movie and its title character fit in to the MCU. References were made to three different cinematic Spider-Man continuities, as well as the Venom movies. Almost all of those Easter eggs were dropped in the finished film. In the end, Morbius takes place in the same universe as Venom and Venom: Let There be Carnage.
But wait, if that’s the case, then why is Adriam Toomes, a character from the MCU Spider-Man movies appearing in Venom?
The answer to that comes in Spider-Man: No Way Home, the third MCU solo Spider-Man movie. As part of a magic spell that went wrong, various villains from previous Spider-Man movie continuities, as well as the Spider-Men themselves, were brought into the MCU for a couple of days. Oh, and Venom was there too, but as we learned in the No Way Home post-credits scenes, he just got drunk. By the end of the movie, via magic again, all the non-MCU characters were sent to their home worlds.
The same magical, sky-tearing effect from the climax of Spider-Man: No Way Home was shown in the Morbius mid-credits scene. Apparently, as the rules of the spells made little sense in No Way Home’s own narrative, the second spell caused the Vulture to teleport to the world of Venom and Morbius. It was probably linked to him knowing Spider-Man’s secret identity or something, but don’t think too hard about it. The writers clearly didn’t.
Now Vulture is stranded in the Venomverse and for some unexplained reason is working with Morbius. It’s just as well, since his plot thread in the MCU doesn’t look like it was ever going to lead to anything. Considering Sony wants to build up a continuity made up of Spider-Man-related characters in a world that may or may not have its own Spider-Man, it appears that they’re trying to build up to a Sinister Six team-up movie much like how the MCU made a ton of money with Avengers.
In the end, this means that Morbius is not part of the MCU itself, but a neighboring universe. It’s connected to it. But given how the MCU is using the multiverse as a plot device in various TV shows and movies (hell, the current era of Marvel projects has officially been dubbed “The Multiverse Saga”), one could make the argument that Morbius is indeed part of the MCU, just not in the 616 universe of the big screen. Your mileage, of course, may vary.