Well known for its use in Gershwin’s ‘An American in Paris’, I realized today I had no idea what in fact a ‘taxi horn’ was! If you Google it you’ll just get references to Gershwin, especially since the discovery that ‘we were doing it wrong’. (See my post here.) Consider that the horns Gershwin used were NOT what you’d typically find installed on an automobile by the manufacturer. (See vintage auto ‘bulb horns’, including many from French manufacturers here.) So what were they?
I turned to Google Books, sorted by date, and looked for references to ‘taxi horns’ from the early 1900s. Gershwin wrote ‘An American in Paris’ in 1928, and the thinking behind using taxi horns in the composition was to evoke the ambience of Paris.
Here’s what I think… In the 1920s, taxi drivers, especially in France, would sound a ‘taxi horn’ when arriving to pick up their fare.
The ‘rattle of wheels’ sounds to me like it could be a carriage.
The Man Thou Gavest
By Harriet Theresa Comstock · 1917
Theatre Magazine Volume 36
Published: 1922
Publisher: Theatre Magazine Company
About now (1940) the term is becoming ambiguous. This could certainly be a taxi’s horn, rather than a ‘taxi horn’. 
But maybe you know better, or have something to add. Feel free to leave us a comment. Thanks!