I think it just sort of happened as a mechanism of the curse. I can’t see EQ sitting down with a chart and making up new names for people. (And if she did, I would think they would be mean names. Snow would have been Jezebel or Slush or something.) I would just see them waking up with knowledge of their names, just as they knew where they lived and what jobs they had.
Sort of like the ‘Twilight Zone.’ But that’s a funny image of the Evil Queen coming up with names for them. Calling Snow White ‘Slushy’ which would be a good name for her if she were a dwarf but there are no female dwarfs.