Can y'all show me what I am doing wrong when i run the package is shows that it is using the $name from the Cat package and not Dog

Read our.

package Dog;
our $name="Odie";
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Here you are creating an alias of the package variable $Dog::name whose scope goes from there to end of file and spans packages. So, in your package Cat the variable $name also refers to $Dog::name whose value you are overwriting.

package Dog;
our $name = "Odie";
print "dog's name is '$name'\n";

package Cat;
$name = 'Garfield';
print "package var \$Cat::name is '$Cat::name'\n";
print "\$name is '$name'\n";
print "package var \$Dog::name is '$Dog::name'\n";
__END__
dog's name is 'Odie'
package var $Cat::name is ''
$name is 'Garfield'
package var $Dog::name is 'Garfield'
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When you call Dog::play ($name); you are not passing the idea of a variable called $name to Dog::play, but the contents the the variable in current scope. You call Dog::play ($name); in the Cat package, and it has variable called $name. So Perl inserts Garfield into the argument list.

Of course, this is further confused by your use of our and lack of scope declaration for $name = "Garfield";. If instead you write our $name = "Garfield";, you can now get the two different results by calling

Dog::play ($Dog::name); # Prints Odie catches a tennis ball.
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Dog::play ($name); # Prints Garfield catches a tennis ball.
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The comments about better methodology would have helped catch and clarify these issues a number of different ways.

package Animal;
use Moo;
has [qw(name gender)] => (is => 'ro');

package Pet;
use Moo::Role;
has [qw(treats pref)] => (is => 'ro');
sub play { sprintf "%s a %s", $_[0]->pref, @{$_[0]->treats}[rand@{$_[0
+]->treats}] }

package Dog;
use Moo; extends 'Animal'; with 'Pet';

package Cat;
use Moo; extends 'Animal'; with 'Pet';

package main;
my @pets = (
    Dog->new( name => 'Odie', gender => 'male', pref => 'chases', trea
+ts => [qw(ball bone doll)] ),
    Cat->new( name => 'Garfield', gender => 'male', pref => 'eats', tr
+eats => [qw(lasagna pookie blanket)] ),
);

printf "%s, being %s, %s.\n", $_->name, $_->gender, $_->play for @pets
+;
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package Dog;
our $name="Odie";
our $gender = "male";
@treats = ( "lasaggnah", "3 cheese pizzas", "taco shells", "BRAAPP" );
sub play{
    my $dogname = shift;
    print "$dogname catches a tennis ball.\n";
}

package Cat;
my $name = "Garmfield";
print "Treats are : @Dog::treats \n";
print "$name says hello to $Dog::name. \n" ;
print "$Dog::name is a $Dog::gender. \n" ;
print "$name prefers kicking $Dog::name.\n" ;
print "$Dog::name loves ",join (" , ", @Dog::treats) ,
    " but prefers $Dog::treats[0]\n" ;
Dog::play ($name);
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package Dog {
   our $name="Odie";
}
package Cat {
   $name = "Garfield";
   print "$name says hello to $Dog::name. \n" ;
}
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