It is called the Doppler effect because it is an effect proposed by
Doppler, in his
Ueber das farbige Licht der Doppelsterne und einiger andere Gestirne des Himmels (About the coloured light of double stars and other heavenly bodies). He assumed that all the colour differences of stars were caused by their movement towards or away from us.
It was the Dutch scientist
Buys Ballot who realised that the effect would also apply to sound, and he set out an experiment where people with perfect pitch would try to determine the change in pitch of notes played by musicians on a train driving by. An experiment that proved very difficult as the notes played could be very difficult to hear because of the sound of the locomotive.