This has basically been answered already, but it was confirmed in the BoB discord by Orion that not remembering their FF past isn't a rule, it's strongly encouraged. The reason staff doesn't want any names mentioned is pretty much just for 'copyright' problems, y'know? Summary: it's not a rule but it's strongly encouraged that your character doesn't remember any past names of FF groups/characters.

[TW: Slight mentions of torture near the end of this paragraph]
For me, I did similar to Cobs - Caera was a leader of The Cartel and went through a lot of difficult things that changed her entire character. I wouldn't just be able to forget all that and go back to her original character, because that's just cheap after developing her for so long, so I gave her a slight case of amnesia (if that's the right word for it, I'm not sure). Caera can faintly remember the basic details of what happened to her (father abandoning her, being captured and tortured, ending up in charge of a group she felt didn't actually like her) but can't remember the small details (names, faces, territory details, etc.). I figured this would be a cool development for her too - she\s always been hung up on her past, so suddenly ending up in a new world after having died on FF with no memories of her past is going to be a challenge for her and will force her to change in order to adapt to her new life.
Of course, as others stated, you can completely start your character over or even have your character remember their past if you absoloutely can't find a way to play your character without that information. It's all up to you - if you're uncertain about things, you can always message one of the staff directly (they're all very friendly, I promise) just make absolute sure, post a thread like this one, or you can even message me/a site friend!

Good luck with figuring out how to place your character here. <3