The initialization of $self is wrong as well. You probably wanted to write:

my $self = {
  farmName => '',
  farmMammals => [],
  farmBirds => []
}
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Usage of => improves readability and allows you to omit apostrophes on LHS
$ in $farmName makes no sense - only if you wanted a hash item indexed by the value of variable $farmName
() is an empty list, inside a list it's the same as if it weren't there at all. [] is an empty arrayref, that's what you need.