On March 9, 1898, a company in Paris started offering rides by motorized cabriolets that included rudimentary taximeters measuring distance. They metonymically called these vehicles taxametres, which they switched to taximetres by 1904. By early 1907, services in London and New York began to pick up on the name, and marketed their rides as either taximeters or taximeter cabriolets. However, because those words were so clunky, they were shortened to taxi and taxicab, respectively (the latter was further clipped to cab, as I mentioned in my post two days ago). The terms rose concurrently with the rise of the automobile, reaching mainstream use by the late 1910s and increasing in usage since then. The word taximeter is basically equivalent to tax and meter, so that's pretty self-explanatory.
Adam Aleksic, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.