Declare @match before the loop (so that it is not overwritten each time), append any matches to @match, and move the print statement outside the while loop.

#!/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
[download]

3 matches found: second_string, first_string, third_string,
[download]

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

Note that the var $string_found and the braces around the regex are superfluous

How's this? I used (?:) to shorten the regex; and parens in the join to get rid of the dangling comma.

use strict;
use warnings;

my @matches;
while (<DATA>) {
   push @matches, $1 if /((?:first|second|third|fourth|fifth)_string)/
+;
}
print scalar @matches,
      " matches found: ",
      join(", ", @matches),
      "\n";
[download]

Nice try, but what will happen if you have more than one string matching in one line? You will only catch the first one and miss all subsequent ones.

CountZero

A program should be light and agile, its subroutines connected like a string of pearls. The spirit and intent of the program should be retained throughout. There should be neither too little or too much, neither needless loops nor useless variables, neither lack of structure nor overwhelming rigidity." - The Tao of Programming, 4.1 - Geoffrey James

I wasn't exactly sure what you wanted as there is some ambiguity in the spec. I interpreted the intent to be: number of the strings that matched anywhere (not total number of matches of those strings) and which ones were they? Code below can be tweaked to do different things.

A few points that you may find interesting is that
1.To read the entire file into a single Perl variable, no while{} loop is required, just undef the record separator and do a scalar assignment! BTW, you will often see @var=(),$var=() and this has same effect as =undef.
2. It is possible to increment a hash value that doesn't even exist yet! Perl will create it and set value =1 on the first increment.
3. You will see that I "cheated" in the print. The concatenation op like: "".keys(%seen) forces keys into a scalar context, scalar keys(%seen) would have been fine too, but this is just a fine point that you may find useful later, eg, print "blah".@array is different than print "blah", @array.
4. I took the /m out of the regex, you may need this if a string spans across \n boundaries.
5. You can expand this code to say print a table of number of times each string matched, etc.

#!/usr/bin/perl -w
use strict; 

my %seen;           

$/=undef;        #input record separator doesn't matter any more
my $data = <DATA>;  #whole file is read into one variable!

my @match = ( $data =~ /(first_string|second_string|third_string|fourt
+h_string|fifth_string)/g );
foreach (@match)
{
  $seen{$_}++;  #yes, you can increment a hash key that
                #doesn't exit yet!
} 
#now we have a hash of strings that were seen.
#also know the number times each string was seen.
#but that doesn't appear to be necessary to know that here?

print "".keys(%seen)," matches found: ", join(", ",sort keys(%seen)),"
+\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

__END__

__above prints:
3 matches found: first_string, second_string, third_string
[download]

Since you don't seem to be using the captures there's no point in remembering them so just increment the count each time.

use strict;
use warnings;

my $matchCount = 0;
while( <DATA> )
{
    $matchCount += () = m{(
       first_string
       |
       second_string
       |
       third_string
       |
       fourth_string
       |
       fifth_string)}xg;
}
print qq{$matchCount matches found\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
[download]

Produces.

3 matches found
[download]

I hope this is of interest.

Cheers,

JohnGG

Update: Oops, misread the OP, you are using the captures so ignore this.

Update 2: Code amended to keep the captures.

use strict;
use warnings;

my @matches    = ();
my $matchCount = 0;
while( <DATA> )
{
    $matchCount += my( @caps ) = m{(
       first_string
       |
       second_string
       |
       third_string
       |
       fourth_string
       |
       fifth_string)}xg;
    push @matches, @caps;
}
print qq{$matchCount matches found: @matches\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
[download]

3 matches found: second_string first_string third_string
[download]