How about "The soul of discretion"? Promise you won't tell anyone? I promise! I'm the soul of discretion.

Tight lipped.

He likes to keep to himself He's not much of a talker He's an introvert

Thank you! :)

! A true friend. A Sphinx. A confidante. Maybe a clam. Often you'd recast this to use an adjective like close-mouthed, tight-lipped, reliable, discreet.