I suspect that that secretary or whatever with the arrowhead in "The Menagerie" was just a costuming blunder that fell through the cracks, and even if someone along the line caught it, it wasn't considered important enough to fix and/or they decided no viewer would notice or care.
No doubt. But in-universe... Suppose Starfleet is divided into "fleet" and "base," or "space" and "ground, " or "fleet operations" and "fleet support," or whatever you prefer ("outpost" might be a further category). "Fleet" personnel wear the insignia of their unit (fleet, squadron, command, whatever you prefer), "base" wear the flower badge. But starbases support operating units, so it would be understandable if there was some overlap of "fleet" personnel working at the base, even though they might ultimately report to a different command than the "base" personnel.