Assignment patch

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602 Club patches large — Several patches on display in the 602 Club

Commodore Robert Wesley wears the "Starfleet Command" assignment patch

An assignment patch was a patch or badge seen on uniforms from Earth starting in the 20th century. The United Earth Starfleet and MACO personal continued to use assignment patches to denote the ship, station, or other post to which an individual was assigned to, as did the latter Federation Starfleet.

History

In the 20th and 21st centuries, assignment patches were used on Military uniforms and civilian uniforms alike. NASA patches from this time period could be seen in the bar at Zefram Cochrane's missile complex in Bozeman, Montana and at the 602 Club in San Francisco. {Star Trek: First Contact, ENT: "First Flight")

While most of the patches seen at the 602 Club were real NASA mission patches, one case contained fictional patches from the Star Trek universe, many of them seen for the first and only time in this episode.

In the early 22nd century, Starfleet personnel wore assignment patches on the sleeve of their left arm hearkening back to the first days of space flight. The uniform would later be updated adding the Starfleet patch to the right arm. (ENT)

In the mid-23rd century, Starfleet continued the tradition of maintaining different assignment patch emblems for individual posts, now worn over the left breast. By 2278, however, this practice was abolished and Starfleet adopted the USS Enterprise's assignment patch as the standard emblem for all Starfleet personnel. At this point, assignment patches fell into disuse in Starfleet, as they began using badges and combadges. (TOS, TNG: "Cause and Effect")

Starfleet Academy would bring the use of an assignment patch back in the 24th century with the introduction of the Red Squad unit in 2372. (DS9: "Homefront")

In the mirror universe, the 22nd century tradition was upheld. While the Terran Empire's uniforms had several variations from that of the prime universe, they still wore mission patches on their shoulders. By the 23rd century though, the mission patch was already out of use in favor of the Imperial insignia. (ENT: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II", TOS: "Mirror, Mirror")

Background

The last assignment patch seen in the 23rd century was the Epsilon IX Station in 2272. (Star Trek: The Motion Picture)
The Starfleet pins adopting the Enterprise emblem were first seen in Star Trek II: The Wrath of Khan; this badge was made into a combadge by 2344, the era of TNG: "Yesterday's Enterprise". Newer combadges were first seen in TNG: "Encounter at Farpoint".
The assignment patch for the USS Enterprise (NCC-1701) and the movie era Starfleet assignment badge are very similar to those used by US Space Command in the 20th century and early 21st century.
There are some stray elements which detract from the "one ship, one emblem" theory, however; some members of Kirk's academy class who cajoled Kirk in a lounge in TOS: "Court Martial" also wore the Enterprise patch. It would seem unlikely that Kirk's direct subordinates would take part in such a thing, classmate or no, so it least suggests they were assigned elsewhere, yet had the same patch.
In the alternate reality created by Nero destroying the USS Kelvin, assignment patches were done away with much sooner than in the prime timeline, with the Kelvin's patch becoming Starfleet's standard by the 2250s. (Star Trek)