A skeleton would be immune to physic damage because a skeleton is not alive and cannot think or process any information.
So they would be unaffected from the physic damage thrust upon it.
However a skeleton can be physically damaged but not physic damaged.
No creature is immune to any kind of damage unless its stat block explicitly says it is.
You can think of psychic damage as interfering with a creature's "nervous system" or whatever magical thing it uses as an equivalent.
Can an undead creature, such as a vampire, take psychic damage from a mystic?
If the creature does not have resistance/immunity to psychic damage in the stat block, it can.
A creature is immune to Psychic Damage, Fear, and Charm.
If it gets attacked by Dissonant Whispers or Mind Sliver do you have it suffer the secondary effects or no because it is immune to psychic damage? Yes, absolutely.
Same reason a creature immune to poison damage is not immune to non-damage effects from a poison.