In present-day navies we see that they have ceremonies and recognition of events like when a ship crosses the equator, or the international dateline, and so forth. Not really official, just a fun tradition that has become part of the naval tradition and culture.
What sorts of traditions and culture do we imagine have established themselves in Starfleet? We've gotten small tastes of Starfleet slang (i.e. "Captain dunsel"). It's a vast starfaring organization that's been around for awhile by Kirk's era, and surely they have something like that going on down on the lower decks?
For example: In the early days of the exploration and colonization of the solar system (before interstellar travel), members of the UN Space Force would get a distinctive tattoo ringing their upper arm when they had been on a mission that circumnavigated the Sun. Many young recruits would end up buying a round of drinks when they failed a "ring check" in the favorite UNSF watering holes on Luna.