You have two basic problems here.

The first problem, the uninitialized errors, is that your new class method is not actually creating the object, or at least isn't keeping it. You're calling register, but you're discarding the registered object. I'd suggest:

# object constructor
sub new {
   my $self = register( shift );
   $given_name{ id $self } = shift;
   $surname   { id $self } = shift;
   $self
}
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The second problem is that your calls to the readonly function, and it is a function!, happens after you're done trying to print $person->given_name - in other words, Class::InsideOut doesn't know anything about these readonly attributes yet! If you were to put your package into My/Person.pm so you could use it, that would solve your problem. Or, for test purposes, move it to the top of the file so it gets executed first.

With those two changes, and an extra space after Name is just so it looks nicer, I have:

#!/usr/bin/perl

package My::Person;

use strict;
use warnings;
use Class::InsideOut qw[public readonly private register id];

# declare the attributes
readonly  given_name   => my %given_name;
readonly  surname      => my %surname;

# object constructor
sub new {
   my $self = register( shift );
   $given_name{ id $self } = shift;
   $surname   { id $self } = shift;
   $self
}

package main;

use strict;
use warnings;

my $person = My::Person->new( 'neil', 'watson' );
print 'Name is ', $person->given_name();
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Good luck!