In your example $name will contain the value of the last line of file1 only, as you overwrite it on each pass through the loop. I suspect you want all the names in file1, so you can store them in a hash instead. Also, if ($element = $name) will always be true, what you need is if ($element eq $name). If I am right about you wanting to check all file1 names then you could do:

use strict;
use warnings;
use Data::Dumper;

my $file1 = $ARGV[0];
my $file2 = $ARGV[1];
my %names;

open( my $fh1, '<', $file1 ) or die "Cannot open $file1: $!";
while (my $line = <$fh1>)  {
    my @data = split(" ", $line);
    $names{ $data[0] }++;
}
close( $fh1 );

# have a look to see what is in %names
print Dumper( \%names );

open( my $fh2, '<', $file2 ) or die "Cannot open $file2: $!";
while (my $line = <$fh2>)  {
    my @data2 = split(" ", $line);
    my $element = $data2[4];
    if ( $names{ $element } ) {
        print join("\t", @data2 ), "\n";
    }
}
close( $fh2 );
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