If you need an in-universe explanation, you could assume that up until a year or so before the Enterprise's five-year mission began, every starship did have its own emblem. Then Starfleet ordered that every ship use the delta. To make the transition easy, since they didn't have replicators then, they would change at the end of their mission during the refit period. The Enterprise just got hers, so they wore deltas; other Starfleet ships might or might not be using the delta depending on how long they had been on their missions.
That's a very plausible in-universe explanation!
