fill is a combination of pattern and 2 colors so for a solid pattern the color you want is probably in the first color
$format->{Fill}->[1]. You also need to create a new format object for each color rather that re-using the same one with set_bg_color

Try this

#!perl
use strict;
use Data::Dumper;

use Spreadsheet::WriteExcel;
use Spreadsheet::ParseExcel;

my $parser          = Spreadsheet::ParseExcel->new();
my $workbook_parse  = $parser->parse( 'Report.xls' );

my $worksheet_parse = $workbook_parse->Worksheet( "Health_Report" );
my ( $col_min, $col_max ) = $worksheet_parse->col_range();
my ( $row_min, $row_max ) = $worksheet_parse->row_range();

my $workbook   = Spreadsheet::WriteExcel->new( "Report_new.xls" );
my $worksheet  = $workbook->addworksheet( "Health_Report" );
my %bkgd_color = ();

for my $col ( $col_min .. $col_max ) {

    for my $row ( $row_min .. $row_max ) {

        # Return the cell object at $row and $col

        my $cell = $worksheet_parse->get_cell( $row, $col );
        next unless $cell;

        my $value           = $cell->value();
        my $format          = $cell->get_format();
         
        my $pattern = $format->{Fill}->[0];         
        my $color1  = $format->{Fill}->[1];
        my $color2  = $format->{Fill}->[2];

        print "Row, Col = ($row, $col) ";
        print "Value    = $value\n";
        print "Fill     = $pattern $color1 $color2\n";

        ### Here trying to rewrite into Excel and apply the
        ### same background color which the cell had previously
       
        if ($pattern == 1 ){
          if ( ! exists $bkgd_color{$color1} ){
            $bkgd_color{$color1} =  $workbook->addformat(
              pattern  => $pattern,
              bg_color => $color1
            );
          } 
          $worksheet->write( $row, $col, $value, $bkgd_color{$color1})
        } else {
          $worksheet->write( $row, $col, $value);
        }        
    }
}
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