The first mention of the structure is attributed to Herophilos of Chalkedon, a Greek doctor who taught in Alexandria around 300 BC. Further descriptions followed, but it was Gabriele Falloppio (1523-1562) who extended the anatomical knowledge of the structure to such an extent that it still bears his name in many countries (including the Latin tuba Fallopii, English Fallopian tube). In Spanish it is Trompa uterina.
But older names for the structure included the egg duct, mother's trumpet, cornu (uteri), cornu matricis, humerus, keraía, latera, ligamentum cornuale, oviductus, ovarialtube, vas seminale, vas spermaticum, vena tenuissima and via medulla.
As can be seen from some of these that Fallopio's contribution to the knowledge about that structure was significant, but the naming was certainly done with 'prior art' in mind.
The cornu was an ancient musical instrument that had different shapes over time:
And older trumpets also had not always the complicated look one sees today. The variations were quite manifold across the world and times, but the ancient Roman tuba is drawn on Wikipedia like this:
So while Fallppio is apparently on record with naming it explicitly after a musical instrument, lets keep in mind what that instrument really is:
Anything that is hollow and cylindrical in shape.?
That would be the definition of tube in English synonmous with pipe. The same in German were English tube becomes Rohr or Röhre. Medical terms are often somewhat more profane in Germany so the Falloppian tubes becomes Eileiter (from Oviductus) but in the ears the Eustachii tubes are the Eustach'sche Röhren.
More on the history and history of the names in
Robert Herrlinger & Edith Feiner: "Why Did Vesalius Not Discover The Fallopian Tubes?", Med Hist. 1964 Oct; 8(4): 335–341. PMCID: PMC1033409
PMID: 14230138. (PDF) (Curiously arguing that Vesalius in fact did describe them pretty detailed and accurately. Only to be hampered by Galenic thought and doctrine.)