The problem is earlier where $id is set for the > lines but then cleared on the subsequent sequence lines

while(<$fh>){
  my $id = '';
  chomp;
  if ($_ =~ /^>(.+)/){
    $id = $1;
  } else {
    $id2seq{$id} .= $_;
  }
}
[download]

Try

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

my $id;
my %id2seq;

my $infile = 'human_hg19_circRNAs_putative_spliced_sequence.fa';
open my $fh,'<',$infile 
     or die "Could not open $infile : $!";
while (<$fh>){
  if ( /^>(.+)/ ){
    $id = (split /\|/, $1)[0];
  }
  $id2seq{$id} .= $_;
}

foreach my $id (keys %id2seq){
  my $filename = $id.'.fa';
  print "Creating $filename\n";
  open my $out_fh,'>', $filename 
       or die "Could not open $filename : $!";
  print $out_fh $id2seq{$id};
  close $out_fh;
}

close $fh;
[download]