We've all seen this arrowhead thing and generally associate it with Starfleet but it wasn't originally a Starfleet wide insignia. If you watched the original Star Trek series and paid attention, you noticed it was specifically an Enterprise crew insignia.
The Enterprise and her sister ships each had their own assignment patches much the same way military ships and units do today. Off the top of me head I can think of the Constellation's patch from "The Doomsday Machine" and the Exeter's patch from "The Omega Glory."
Starfleet Command had its own patch as well.
But by the time of the movies, the Enterprise patch has been adopted Starfleet wide. The Enterprise under Kirk's command had one of the most successful, if not the most successful, five year missions. But is that enough for Starfleet to make that ship's patch the insignia of Starfleet? Since Starfleet did the answer must be yes.
I understand the real world reasons why. It simplifies wardrobe, prop, and continuity issues if everyone is using the same thing. But it's still something that has never completely sat well with me since a distinctive ship insignia is a powerful symbol of pride (unless you hate your assignment of course).