"Omni" means "All" in Greek; "Glot" means "Having as tongue", also in Greek.

Anatidaephobia is defined as a pervasive, irrational fear that one is being watched by a duck. The anatidaephobic individual fears that no matter where they are or what they are doing, a duck watches.

Anatidaephobia is derived from the Greek word "anatidae", meaning ducks, geese or swans and "phobos" meaning fear.

An encyclopedia of all languages

Omni; possessing all of its class, glot; 'to have a tounge'

So, possessing all tounges, but in this case it is languages.

Also see omniglot.com