I am wondering if any monsters have a hypnosis-like spell or feature, and if so what is the ability used in the saving throw?
What I mean about "hypnosis" is that it falls short of complete mind control. I.E. they person conducting the hypnosis can control the person except they cannot make them kill themselves, and every time they are damaged the can repeat the saving throw ending the hypnosis on a success and cannot he hypnotized by the same person within 24hrs.
Best Answer
It's up to the DM, but there are a few examples of monsters with hypnosis abilities we can draw guidance from.
The Kelpie (From White Plume Mountain in Tales from the Yawning Portal) has an ability called Drowning Hypnosis:
The Morkoth (VGtM, pg. 178) has an ability called Hypnosis:
The Ultroloth's (MM, pg. 314) ability Hypnotic Gaze reads:
The Demogorgon from Out of the Abyss has a hypnotic gaze ability:
A Krasis from Guildmaster's Guide to Ravnica can have the Hypnotic Display ability:
The precedent seems to be Wisdom, but the DM can choose otherwise if they have good reason to. The Kelpie's ability description is quite long, I'd prefer not to reproduce it here in its entirety (staying abreast of guidance in this meta answer), but the Morkoth's ability is more brief and gives a good template for what hypnosis could be.
[This section was made less relevant due to an edit to the question] Finally, there is also the crowd control spell, hypnotic pattern: