I never really hear about Greek folklore/mythical creatures outside of the Ancient eras. I only ever hear of western European folklore most of the time when it comes to the medieval ages and so forth.
Depends on what you mean. We do see mythical creatures such as griffin in crests and all and we do see dragons in Greece too especially in association to saints. For example Saint George is being shown killing dragons before. It is the same logic as in the west quite frankly.
Ancient mythical creatures or deities were also used as euphemisms or symbolisms in alchemy and alchemical texts such as Hermes/Mercury being the ultimate form of their mysticism as the knowledge-seeker and the resurrection symbol. The deities were also associated with their respected planets through the astrological symbols of the mysticism (Aphrodite-Venus, Ares-Mars etc)
Ancient symbols are also still used in modern day era such as Asclepius chalice and snake for pharmacies or Hermes's staff for merchants. We of course have the myths of sea creatures like mermaids (taking the lead from ancient Triton) that appear in art or in folk songs.
Myths are also used in everyday speech even to this day. Terms such as "Achilles's heel" when someone speaks on a weak spot or in Greek for instance we don't say "out of the frying pan and into the fire" to speak that someone went from one bad situation to another even worse. We say "από τη Σκύλλα στη Χάρυβδη" aka "from Skylla to Charybdis" referencing of course the Odyssey or when someone says they narrowly escaped a situation that would be costly we say "πέρασε τις Συμπληγάδες" aka "he passed through the clashing stones" referencing the Argonauts. When someone constantly speaks on bad things that they fear will happen in the future we often call that fellow "Cassandra" referencing to that or when someone sacrifices themselves or is taken down as a scapegoat we often refer to them as "Iphigenia"
We also use the ancient phrases of the poems as proverbs. For example Hector's phrase "there is no better omen than fighting for one's country" or "fear of the Danaans even if they bear gifts" etc.
Our folklore is basically totally merged to our everyday life at this point or even in older times