Why an array has the name names_hash? You are correct you want hash there, so define hash, not array:

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

my $names;
my @name;
while(<DATA>)
{
    if(/{(.*)}/)
    {
        $names = $1;
        my %names_hash = qw{NAME 1 AGE 1};
        push @name, $names;
        
        next if ! exists $names_hash{$names};
        print $names;
    }
}
print @name;
__DATA__
{NAME}
{AGE}
{SEX}
{ADDRESS}
{ADDRESS}
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