In this case, you want all of the DATA in one go rather than in six separate lines. To achieve this, it'd be easiest to simply localize the input record separator $/, see perlvar for details. Basically, asking for <DATA> in a while loop grabs one record at a time, in this case one line at a time. That's because the input record separator is newline by default. If you do a local $/, you're effectively removing record separations, and the whole DATA segment will be read in at one time.

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

while (<DATA>) {
my $string_found = $_;
my @match = ( $string_found =~ /(first_string|second_string|third_stri
+ng|fourth_string|fifth_string)/gm );
print scalar @match," matches found: ",join ", ",@match, "\n";
}

__DATA__
This is a test file matchme ljldjlfjd l;djfldjlf d
test test test dljfldjlfjldjfldjlljdf one second_string
dlfjldfj  ljdfldjjf ldjfljdl dfljdlfj dfdlfj
three ljfldjlj dlfjlasdj foiidufoiida matchdf
dljfldsaofuoidfousdaof ladsjflasdof first_string
dlfjodsuofuasdo sadoufosadu foasduf aosduf third_string
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Produces this result:

3 matches found: second_string, first_string, third_string,
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