You mean something like this?

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

my $str = '"$a $a $a" ab bc "c$d e$f$" c$d "$a $a $a"';

my @uniq =
    keys %{{map {$_ => 1}($str=~m/("[^"]*\$[^"]*")/g)}};

print Dumper(\@uniq);
[download]

And the output is -

$VAR1 = [
          '"c$d e$f$"',
          '"$a $a $a"'
        ];
[download]

Note that I kept the regex simple by not trying to escape the string in the example above. If you want to include escaped strings, the regex becomes more complicated.

...
my $str = '"$a $a $a" ab bc "c$d \"e$f$" c$d "$a $a $a"';

my @uniq =
    keys %{{map {$_ => 1}
      ($str=~m/("(?:\\"|.)*?\$(?:\\"|.)*?")/g)}};
...
[download]

And the new output -

$VAR1 = [
          '"c$d \\"e$f$"',
          '"$a $a $a"'
        ];
[download]