if ($words =~ /"{2,}/g) { ... }

if ($words =~ /"{1,}/g) { ... }

if ($words =~ /"[^"]+"/) {
    my $phrase;
    my $otherwords;
    ($phrase, $otherwords) = extract_quotelike($words);
    
    $phrase =~ s/"//g; #Anführungszeichen entfernen
    @splitwords = split(/\s+/, $otherwords);
    push (@splitwords, $phrase);
}

#!/usr/bin/perl

use strict;
use warnings;
use Text::Balanced qw(extract_delimited);
use Data::Dumper;

my @strings = (q~Angela "Königin von Berlin" Merkel~, 
               q~Angela "Königin von Berlin"~,
               q~"Königin von Berlin" Merkel~);

for my $text(@strings){
    my @result = extract_delimited($text,'"','.*?');
    print Dumper(\@result);
}

if ('"' eq substr($word, 0, 1) && '"' eq substr($word, -1, 1)) {
...
}

use strict;
use warnings;
use Data::Dumper;

my @strings = (
   'Die Angela "Koenigin von Berlin" Merkel ...',
   'Die Angela "Koenigin von Berlin" ...',
   'Die "Koenigin von Berlin" Merkel ...',
   'Die "Angela" Koenigin von Berlin "Merkel" ...'
);

print "--------> BEISPIEL 1\n";

# wenn du "Die Angela" ebenfalls durch Whitespaces trennen moechtest
for my $text (@strings) {
   my @result;
   for my $splitted (split /(".+?")/, $text) {
      if ($splitted =~ /^".+"$/) {
         $splitted =~ s/"//g;
         push @result, $splitted;
      } else {
         push @result, split /\s+/, $splitted;
      }
   }
   print Dumper(\@result);
}

print "\n--------> BEISPIEL 2\n";

# wenn du "Die Angela" nicht durch Whitespaces trennen moechtest
for my $text (@strings) {
   my @result;
   for my $splitted (split /(".+?")/, $text) {
      $splitted =~ s/"//g;
      push @result, $splitted;
   }
   print Dumper(\@result);
}