A variation: If the data file is large, you may want to avoid slurping it all into memory. In which case, set the input record separator $/ to a suitable string:

#! perl
use strict;
use warnings;
use feature 'say';

{
    local $/ = 'TLE - Tag Logical Element';

    while (<DATA>)
    {
        say "$1 $2" if /FQName\s+'(.*?)'.*\|-\s+'(.*?)'/s;
    }
}

__DATA__
<as before>
[download]

(Regex based on hdb’s ++answer.) Output:

12:17 >perl 802_SoPW.pl
ENVELOPEID 5172986
CUSTOMER_NAME Acme Inc
KUNDNAVN BANANA AIRLINES
ZIP_CODE SE-971 74
COUNTRY SE
FAKTURA 5643452
KONTRAKT 90234342
REGISTER AF89729
DATO 16.12.13

12:17 >
[download]

Hope that helps,