Given the updated OP, a minor mod to my first attempt seems to work?

P:\test>p1
perl> sub merge{
    my( $s1, $s2 ) = @_;
    my $i= length $s1;
    $i-- until substr( $s1, -$i ) eq substr( $s2, 0, $i );
    return $s1 . substr( $s2, $i );
};;
perl> print merge( 'ATTTA',  'TTTAA' );;
ATTTAA
perl> print merge( 'ATGTA', 'ATGTA' );;
ATGTA
perl> print merge( 'ATGATG', 'ATGATG' );;
ATGATG
perl> print merge( 'ATGGTAC', 'CCGTAATG' );;
ATGGTACCGTAATG
[download]

But that still doesn't match the OPs solution to the 3rd example.