The further you stray from the base, the less point is there to base it on a certain franchise. AbullSinCara is not incorrect per se, but if I write a story set in a completely rebuilt world, where I only insist that mutants used to exist, ghouls used to exist, radiation used to be rampant, anarchy used to be common, then what's the point of waving the Fallout-flag?
The point would be to make it recognizeable with the genre and franchise, otherwise it might even come off as false advertisement. The balance between "It's my fan fic! I can write about whatever I want!" and "Dude, I expected at least a raider/ghoul/supermutant/death-claw/NCR-VC-Khan-V101/etc mention in there!"
Personally, not a fan of fan-fic, but I understand the appeal of it.