Hi guys,

I am looking for an official playable race or class that can go toe to toe with the Mindflayers in terms of psychic/psyonic ability.

Other than the Githyanki or Githzerai.

I'm looking for a psychic/psyonic race or class that can match the Mindflayer in terms of mind powers, and which is playable.

I'd also prefer something official if possible

Maybe Kalashtar would be a good fit for what you are wanting?

For just a basic mind flayer it looks like we need the 60-foot cone psychic damage attack, plus access to dominate monster, detect thoughts, and levitate.

I guess the closest thing is a Great Old One warlock, plus take the ascendant step invocation for levitate. You could probably re-flavor a dragonborn's breath weapon to psychic damage. Dominate monster is an 8th-level spell, though. That one takes a pretty high-level caster.

Otherwise, there's no official psionic class. They tested the "Mystic" at one point but pulled it. No word on when WoTC might re-work it.

There are no races the can match illithids because the mind flayers are dedicated psionicists

Official: Kalashtar

Not that much but only one that offers psionic powers with some mental defense.

A Ghostwise Halfling has some telepathy.

Classes: Nothing really but the closest is probably the Great Old One Patron for the Warlock.  In UA there's the Mystic, which is a re-imagining of the 4th Edition's Psionic class, but it's very broken and poorly designed in my opinion. 

4th Edition had more psychic flavour with Psionic class and the Shardmind race but these have not been officially brought into 5th Edition.

You could also do sorcerer with heavy emphasis on psychic damage spells.