It is an analogy made for the wide audience. The actual picture provided by QFT is far more technical.

Configurations of electrons and positrons are **states** of the quantum system, called the electron field. These states can be obtained from the vacuum state by acting with creation operators.

These creation operators evolve in time (in Heisenberg picture), and the their frequency is positive for electrons and negative for positrons (or vice versa, it doesn't matter).

The whole thing is even more complicated in QED, when the interaction affects properties like mass, charge or field normalization factor via the renormalization.