It's a cool badge but, as Digger RJ says, its a license to operate a general hire vehicle registered with the city.
Doctor Edward Miles has a great website for identification of US chauffeur badges etc, listed by state, but unfortunately doesn’t provide any history. At the link below, he illustrates the Worcester Hackney Driver badges in use between 1918-1942:
http://www.chauffeurbadges.com/85.html
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The term “Hackney” in modern times probably originates from the London Borough of Hackney and a “Hackney Carriage” was originally a small two-horse enclosed carriage available for hire and capable of carrying four passengers. London was the first to introduce legislation to regulate their operation, with the first licences issued in 1662.
London’s iconic “Black Cabs” (now also operating in other liveries) are regulated as “Hackney Carriages” and, until the 1960s, despite the absence of any horses the regulations still required them to carry a bale of hay in the boot (trunk).
I don’t know if the same distinction applies (or applied) in the US but, over here in modern times, a Hackney license differs from other types of taxi licensing in that it permits a limited number of approved taxis to operate from the on-street ranks provided by a city or borough (and at airports etc) and allows them to be hailed in the street, as well as on demand or by pre-booking. Other taxi types (in unlimited numbers) are referred to as “private hire” (minicabs, Uber etc) and are only permitted to operate from their own premises or by request to pick up from a given location (whether on demand or by pre-booking).