In a former question it was pointed Mob appears in a cover photo of a One Punch Man manga. But also apparently Tatsumaki appears in a Mob Psycho manga (I've seen the anime but I haven't read the manga), in an apparent scene of the manga itself, not a cover.
Do Mob, Tatsumaki, and Saitama live in the same universe/planet? Kinda like Goku and Arale from Dragon Ball/Dr. Slump?
The answer is NO.
Saitama's world and Mob's world is totally different. We can see that the art style of the anime and the main character is almost alike that's because they were written by the same creator, ONE.
Mob appear in One Punch Man world.
(Image credit to @Pablo.)
Saitama appear in Mob Season 2 Opening. (Watch the cellphone, it's Saitama outfit.)
So why does these 2 not related story will appear on each other scene, I guess it was just a purely an advertisement purpose.
I don’t believe they live in the same universe since in season one episode two someone is seen reading one punch man not to mention saitama appears on the back of a mc Donald’s bag in episode three (or Mob’ Donald’s because copyright) so it seems saitama is just a novelty character in mobs universe
For me, i think that they are in the same universe and i have PROOF as you remember the episode that mob and his sensei reigen fought the ghost in a tunnel in the first season his sensei reigen suddenly got scared and suddenly released his phone quickly and show his phone to the screen with saitama's face to try to call him if something bad happens heres the link of the vid
https://www.youtube.com/watch?v=QsGv6er1fcY Stop the video @ 0:06 seconds and watch the phone
To elaborate on kit's answer, the worlds of Mob Psycho and One Punch Man have incompatible canon. Psychic powers exist in One Punch Man, but other kinds of heroes do not exist as far as we know in Mob Psycho. Additionally, powers are public knowledge in One Punch Man since there is a hero bureau that protects everyone, and there are monster threats and such that cause very large amounts of damage, but psychic powers are a secret to the public in Mob Psycho. We can see this in Reigen's initial reaction to Mob, since he doesn't believe him when he says he has psychic powers. If he lived in One Punch Man's world, then he wouldn't have been so skeptical about the existence of psychic powers, but would have focused his skepticism on whether Mob in particular had them. From what I recall, his reaction was very much of someone who did not believe psychic powers exist at all.
Thus, it is most likely that the appearances of the characters in each others' universes are simply cameos. You can ignore them as far as canon goes, or you can simply assume a different version of Mob exists in One Punch Man. As far as Saitama appearing in Mob Psycho, he could be a fictional character if he only appears on cell phones and stuff. And it's worth noting that Saitama isn't famous in his world, so it doesn't really make sense for people to be using him as their cellphone wallpaper (it's possible to work around this, but due to the unlikelihood, it points to this probably not being the universe of One Punch Man.)
Regarding Arale, she also presents canon inconsistencies to the Dragon Ball universe. She may have meshed better with the original Dragon Ball, but by the time of the Dragon Ball Super episode she appeared in, the tone of the series has changed a lot, to the point where that episode felt a bit like a non sequitur. Vegeta even ends up getting self-referential and meta when he starts calling her a "character from an 80s gag manga". Dragon Ball Z and Dragon Ball Super aren't usually that meta, so it's kind of hard to determine how canonical that episode was, and thus how canonical Arale living in the same world is. The canon of Dragon Ball also presents some problems to Dr. Slump. Has Arale now died at least once, via Majin Buu?
Due to these inconsistencies, we can't really conclude anything definite with regards to the canons of any of these series.
