Lilibet Stern's real name is Elizabeth though. She was no doubt aware of the Queen's nickname and it seems decided, at some point, to use it as a stage name.
I don't think there's any doubt that the name "Lilibet", if indeed it is now deemed to be a name in its own right, originated with Queen Elizabeth II.
It's perfectly possible (even likely) that other little Elizabeths made similar (or even identical) errors in pronunciation of their name as toddlers, of course. But Queen Elizabeth made it famous.