It is possible to knock sheep off of cliffs, but they simply reappear back at the top. You can also "kill" some fantasy creatures in a cartoony way, and some of them resemble real animals.
There are sidequests that involve herding sheep, and you can knock them around with your cap, but they're never implied to be hurt by this. You can also knock the sheep off of cliffs, but they simply reappear back at the top.
If Mario runs out of air while underwater, he starts to lose health, which could be interpreted as him struggling to breathe. He can also die while underwater due to a lack of air, but it's not graphic or realistic, and he respawns shortly afterward. A beta version of the game reportedly showed Mario struggling to breathe in a semi-graphic manner, but this is not present in the final game.
Mario can catch fire (e.g. when he falls in Lava, he bounces off of it while getting burn marks), which can cause Mario to lose all his health, but it's not graphic.
The closest thing in the game is Bowser hanging a cage that has Princess Peach in it, but Peach herself is not hanged. Nobody is hanged in the traditional sense (it's a Mario game, c'mon).
Mario can drown by staying underwater for too long, and he can also die from being sucked into quicksand (while he's not depicted asphyxiating, the implication is there). Mario can also die by turning into an ice statue after staying in cold water for too long, which some might interpret as asphyxiation (though it's supposed to represent him freezing to death). However, none of the deaths are realistic or graphic, and Mario also respawns shortly afterward. Enemies can also die from quicksand, but they respawn after a little while.
Mario teams up with a friendly ghost named Cappy, who accompanies him on his adventure. Cappy spends most of his time living in Mario's hat and grants Mario the ability to possess ("capture") hatless enemies by throwing his hat at them.
In the food kingdom, the bird has a part of the fight that's based around it's vomit i think. you can get enough moons to skip the fight completely, though
A few enemies spit various substances at you (like lava or poison). It doesn't look like real spit (the lava is bright pink and the poison is purple), and both encounters with these enemies aren't required to beat the game.
I don't remember any intentional flashing lights or images in the game itself, but there are reports of a bug on some (unofficial) emulators that cause the screen to rapidly flash upon startup. I don't know if this was patched out of the emulator or not, but if you're playing this on an actual Nintendo Switch then this shouldn't affect you.
Some instances that might be an issue to a few people:
In every kingdom except for the last two, you can buy items at a store called Crazy Cap.
There is a postgame level in the Lost Kingdom where you have to chase after Klepto, who snatches Cappy. When you rescue him, he calls Klepto a “stupid bird.”
After you earn a Power Moon from a Trace-Walking minigame, you can talk to the Koopa again, and he will ask you if you want to give it another try. If you say no, the Koopa says “lame” as part of his response.
Unless you count those as ableist, there's no real ableism in the world of Mario Odyssey.
Kleptomania is stigmatized through the “Klepto” character. Not saying people not in a group should tell people in said group what to get offended by but assuming none of them are offended is the same prospect. I haven’t seen anyone offended by Tostarena’s portrayal or “Mexican Mario,” granted.
Yes, in the Metro Kingdom, Cars will stop when Mario runs in front of them and let out a screech+halt, but it’s relatively quiet, especially with the music.
Nuclear threats don't exist in the game, but there are Bullet Bills, tank enemies called Sherms that fire exploding missiles, and a boss called the RoboBrood that fires bombs from its mouth-shaped cannon.