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]