Briefly, new(...) is a class method whose first parameter is the name of the class. So, what you are calling $self is actually a string rather than an error object. And when you return $self, you are returning the class name string from your constructor rather than the error object.

Your class definition should look more like this: (untested code)

use strict;
use warnings;

package DB::DBException;
use base qw(Error);

sub new {
  my ($sClass, $msg) = @_;
  my @args = ();
  local $Error::Depth = $Error::Depth + 1;
  local $Error::Debug = 1; # Enables storing of stacktrace

  # *** this is the key line change ***
  my $self = $sClass->SUPER::new(-text => $msg, @args);
  return $self;
}
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