If you are just trying to print the value of the 'draw:name' attribute, the use the att method. I had to fix the 1st line of your XML to make it valid.

use strict;
use warnings;
use XML::Twig;

my $xmlStr = <<XML;
<draw:frame draw:id="id2" draw:layer="layout" draw:name="objectName" d
+raw:style-name="gr1" draw:text-style-name="P3">
  <draw:text-box>
    <text:p text:style-name="P1">
      <text:span text:style-name="T2">objectData</text:span>
    </text:p>
  </draw:text-box>
</draw:frame>
XML

my $twig= new XML::Twig(
    twig_handlers => { 'draw:frame' => \&getDrawInfo }
);
$twig->parse($xmlStr);
exit;

sub getDrawInfo {
    my ($t, $data) = @_;
    print $data->att('draw:name'), "\n";
}  

__END__

objectName
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