This isn't the first time we saw multiple Starfleet badges being used at the same time, by different ships. That would be the TOS era, which had about seven different patches worn at the same time.
A Bob Justman memo unearthed a few years ago (by members of The TrekBBS, if I'm not mistaken) revealed that the
Antares patch seen in "Charlie X" was meant to represent the Starfleet merchant marine, and that all capital ships like the
Enterprise were meant to have the arrowhead insignia, as seen in "Court Martial" where everyone in the starbase bar has it. But the makers of season 2 episodes like "The Doomsday Machine" and "The Omega Glory" somehow didn't realize that and gave other ships different insignias, hence the Justman memo coming out subsequently to correct the error. The
Defiant uniforms in "The Tholian Web" go back to the arrowhead, although it was barely visible and the makers of ENT: "In a Mirror, Darkly" missed it, giving that ship a different insignia based on the Starfleet command chevron from TOS.
In my
Enterprise: Rise of the Federation novels, I posited that the different insignias from TOS originally represented the different founder worlds' fleets that combined into the Federation Starfleet -- the arrowhead for UESPA (since VGR: "Friendship One" established it as a UESPA insignia going back to the late 21st century), the
Constellation "pretzel" for Andoria, the hoof-shaped
Antares insignia for Tellar, the rectangular
Exeter insignia for Alpha Centauri, and an IDIC patch for Vulcan. I presumed that over time, as the fleet became more unified, these subfleets evolved into just different administrative subdivisions that used the legacy insignias, so the
Constellation and
Exeter might have been under the administration of those different subdivisions instead of the UESPA division that oversaw the
Enterprise and most other ships we've seen in that era. Eventually, by TMP, they did some kind of reorganization and dropped the other insignias.