(covers information from several alternate timelines)
A 1930 newspaper article quoted motorists without license tags near Lee's Summit using the excuse "I'm going to be drafted next month and don't want to spend $15 or $20." (TOS: "The City on the Edge of Forever")
Usually the alphanumeric sequence is a government sanctioned code that indicated the geographical location or an officially recognized organization a vehicle belonged to, and is an identification aid for law enforcement agencies as well as tax authorities.
When a Star Trek production (or any other motion picture production for that matter) needs a historical, period vehicle, they usually rent these from museums, or more commonly, private collectors. In the late 20th century several of these turned up carrying nonsensical license plates that started with "NA", whether or not followed by any alpha numerical sequence. The "NA" is a code – in everyday parlance the acronym for "not available" or "not applicable" – that indicates the vehicle is not a regular traffic participant (and therefore usually exempt from road taxes), but rather a limited use vehicle only, to be used for example, as exhibition and parade pieces, or as rentals to motion picture productions. Several period vehicles carrying this type of plate have been featured in the Star Trek: Enterprise season four two-part episode "Storm Front", among others. While not officially recognized by authorities, the format has nevertheless been adopted in collector circles of several western nations.