Liv and Maddie are actually one girl with a serious Split Personality issue.We're watching the show from Liv or Maddie's perspective, depending on which girl is real. Either way, she believes her alternate personality is a real person. The other people on the show just play along and pretend Liv/Maddie has a twin sister.
Liv is short for Olivia
Miley Stewart once made herself acquainted to Maddie and thought Liv was an alter ego created by Maddie for similar reasons Miley created Hannah