Time: Varies.
Frequency: Every time? I seem to be able to replicate it every single time.
World name: Famfrit
Character name: Bran Cinaed
NPC name: None
Monster name: None
Class/Level: 50 White Mage
Party or solo: Solo
In-game time: Any
Area and coordinates: Any
Steps:
1. I have chat open with everything enabled except Shout, Yell, and Other Synthesis
2. I mount chocobo
3. The chat pulls a name that isn't mine and says "____ mounts Company Chocobo."

Here are screenshots and how I first noticed it, because monsters cannot mount chocobos!

and here is me doing it multiple time is town showing how it changes names.