The "y" in the word "syringe" is a clue that it comes from Greek. To be specific, English "syringe" comes from Greek
syrinx, which refers to a tube or pipe, including a musical pipe. The word has a story that goes with it: Syrinx (Σύριγξ) was a nymph devoted to Artemis, the virgin goddess of the woods and forest. The god of the woods, Pan, lusted for Syrinx and pursued her; in order to escape, Syrinx prayed to be turned into reeds by the water's edge. Pan cut the reeds and shaped them into a "Pan's pipe," hence the meaning of the word
syrinx, "pipe." You can read more about the nymph Syrinx and her story at
Wikipedia.
You can also read more about pan pipes at
Wikipedia. Here is a fourth-century mosaic of a shepherd with pan pipes: