You were using $family{$label}[$names_idx] as an array ref whether it was an array ref or not. I added an if to avoid the problem.

@names always had one and only one member: a reference to an array of names. You probably meant to do @names = @{$family{$label}};. That you had four nested loops, but only three levels should have been a clue.

0..$#array is harder to read than doing a foreach on the array.

Here are the changes:

foreach my $label ( sort keys(%family) )
{
    print "$label\n";

    foreach my $name ( @{ $family{$label} } )
    {
        print "\t$name\n";
        if (ref($name))
        {
            foreach my $inner_name ( @{ $name } )
            {
                print "\t\t$inner_name\n";
            }
        }
    }
}
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if I use strict, the following code must be removed

That's not a good way of thinking. The code needed to be fixed, not removed. The problem existed whether use strict was used or not. The error was just silent when you removed the use strict.