Works for me. Did you print the results?

#!/usr/bin/perl
use warnings;
use strict;

my $BE_temp = '+--------------+-------------+
| version_name | schema_name |
+--------------+-------------+
| dsmgt_03_03  | dsmgt_03    |
+--------------+-------------+
';

$BE_temp =~ m/^\|\s(\w+_\d+)\s+\|\s+(\w+)\s+\|$/m;
my $LSV = $1;
my $LSN = $2;

print "[$LSV] [$LSN]\n";
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Note that you are filling the variables even if the regex does not match. It might be better to fix that, plus avoid the $1 and $2:

if (my ($LSV, $LSN) = $BE_temp =~ m/^\|\s(\w+_\d+)\s+\|\s+(\w+)\s+\|$/
+m) {
    print "[$LSV] [$LSN]\n";
}
[download]