I wouldn't re-use the same variable name for a lexical that has already been defined. Also, don't put a postfix if on a my, as its behavior is undefined - see the note at the bottom of the section Statement Modifiers. Instead, you could do:

my $self = OO->new( $cat );
{
  my $inner_self = $ARGV[0] eq 'Fido' ? OO->new( $dog ) : $self;
  say $inner_self->species();
}
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Note you shouldn't use Indirect Object Syntax either.

Update: Added some details to the above. Plus, why don't you just do this?

my $self = OO->new( $ARGV[0] eq 'Fido' ? $dog : $cat );
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It's also a bit unusual to use $self as the variable name for a newly created object, since $self is usually used as the first argument to a method call, i.e. it's an existing object.