return SF->new ("$lhs->{str}$rhs->{str}", $lhs->{list}, $rhs->{list});

Shouldn't the merging of the lists look something like

    ...
    return SF->new (
               "$lhs->{str}$rhs->{str}",
               [
                 map ref($_) eq 'ARRAY' ? @$_ : (),
                     $lhs->{list}, $rhs->{list}
               ]
           );
[download]

Another issue (which I don't have a solution for) is that what's LHS and RHS seems to vary with the circumstances of the concatenation, i.e. while this

my $name1 = SF->new ('GrandFather', [1, 2, 3]);
my $name2 = SF->new (' and Almut',  [4, 5, 6]);

my $catStr = $name1 . $name2;

use Data::Dumper;
print Dumper $catStr;
[download]

produces (as expected)

$VAR1 = bless( {
                 'str' => 'GrandFather and Almut',
                 'list' => [
                             1,
                             2,
                             3,
                             4,
                             5,
                             6
                           ]
               }, 'SF' );
[download]

"Name is $name1" (or "Name is " . $name1)  doesn't :

$VAR1 = bless( {
                 'str' => 'GrandFatherName is ',
                 'list' => [
                             1,
                             2,
                             3
                           ]
               }, 'SF' );
[download]

(note that "Name is " is appended to "GrandFather", not the other way round)