I've noticed the Doppler effect in the change of pitch of music as I ride past on my bicycle, so I assume in the old days people did this on horses too. Are there any ancient mentions or theories about this, maybe the ancient greeks?
Have you really performed the experiment with riding a bicycle past a music source? I do not know with what speed you can ride your bicycle, but with a generous assumption of $10$m/sec, I arrive at the frequency distortion of $1.03$. This is less than half-tone (the ratio of the frequencies of two neighboring piano keys (black and white). So I imagine that "an ancient" standing and listening how a horseman riding past him at a full speed sounds a horn will not notice much difference. To have this $1.03$ frequency change, the horseman has to ride straight at you; if he passes you at some distance, the effect will not be sharp but gradual. And the horseman has to produce the same note on his horn, continuously. So I doubt that this was noticed.
By the way Doppler himself discovered the effect by observing light, rather than sound. (Railroads already existed in 1842 when the effect was discovered. One can easily hear the Doppler effect when a train on full speed passes you while sounding the signal).