This Perl code takes in all the to-be merged files and determines the common header from the headers of all the files and then rearranges the column printing order to output the files.

     perl -wMstrict -Mvars='*ARGV_orig,*comm_hdr,*prev,*h' -lne '
     BEGIN{
       @::ARGV_orig = @ARGV;

       $::prev = q//;

       sub trim {
          my ($str) = @_ ? @_ : $_;

          for($str) {
             s/^s*//;s/s*$//;
          }

          return $str;
       }

       sub intersection(@@) {
          @{$_[0]} > @{$_[1]} and @_ = reverse @_;

          my @smaller = @{ +shift };
          my @larger  = @{ +shift };

          my @common;

          for my $e (@smaller) {
             push @common, $e
                if grep { $_ eq $e } @larger;
          }

          return @common;
       }

       sub col_print_order {

          my @common_hdr = @{ $_[0]->{common_header} };
          my @header2prn = @{ $_[0]->{header_2print} };

          my @reorder;

          for my $e (@common_hdr) {
             if ( -1 < (my ($l) = grep { $header2prn[$_] eq $e } 0..$#header2prn) ) {
                push @reorder, $l;
             }
          }

          return @reorder;
       }

    }

    if ( $ARGV ne $::prev ) {
       $::h{$ARGV}{header} = $_;
       my @A = split;
       @::comm_hdr = @::comm_hdr ? intersection(@::comm_hdr, @A) : @A;
       $::prev = $ARGV;
    } else {
       push @{ $::h{$ARGV}{data} }, $_;
    }

    END{
       local $, = chr(32);

       my @comm_hdr_sorted = sort @::comm_hdr;;
       print @comm_hdr_sorted;

       for my $argv (@::ARGV_orig) {
          my @current_header = split /s+/, trim $::h{$argv}{header};

          my @order = col_print_order({
             common_header => @comm_hdr_sorted,
             header_2print => @current_header,
          });

          my @file = @{ $::h{$argv}{data} };

          for my $line_num ( 0..$#file ) {
             my $line = trim $file[$line_num];
             my @fields = split /s+/, $line;
             print @fields[ @order ];
          }
       }
    }
 ' yourfile1 yourfile2 yourfile3 # ... specify all your filenames to be merged here

Output

Col1 Col3 Col4 Col5
A    C    D    E
P    R    Q    S

GNU awk, can handle an arbitrary number of files (all file content is stored in memory, so depends on your system's memory capacity)

gawk '
    # examine the headers for this file
    FNR == 1 {
        num_files++
        delete this_headers
        for (i=1; i<=NF; i++) {
            all_headers[$i]++
            this_headers[i] = $i
        }
        next
    }
    # this is a line of data
    {
        n++
        for (i=1; i<=NF; i++) {
            data[n][this_headers[i]] = $i
        }
    }
    END {
        # find the headers that are common to all files
        for (header in all_headers) {
            if (all_headers[header] == num_files)
                common_headers[header]
        }
        # sort arrays by index, alphabetically
        PROCINFO["sorted_in"] = "@ind_str_asc"
        # print out the common headers
        for (header in common_headers) {
            printf "%s ", header
        }
        print ""
        # print out the data
        for (i=1; i<=n; i++) {
            for (header in common_headers) {
                printf "%s ", data[i][header]
            }
            print ""
        }
    }
' file1 file2

outputs

Col1 Col3 Col4 Col5 
A C D E 
P R Q S 

Save the code as file mergecols, make it executable and start it with mergecols -C1=0,2,3,4 -C2=0,2,1,3 file1 file2

#!/usr/bin/perl -s

# mergecols
# -C1=0,2,3,4   columns from file 1
# -C2=0,2,1,3   columns from file 2
# file1         input file 1
# file2         input file 2

($f1,$f2) = @ARGV;

@t1 = map { [split] } do { local @ARGV=($f1); <> };
@t2 = map { [split] } do { local @ARGV=($f2); <> };

@c1 = split /,/, $C1;
@c2 = split /,/, $C2;

for ( $i=0; $t1[$i] or $t2[$i]; $i++ ) {
   print join ' ', @{$t1[$i]}[@c1], "n" if $t1[$i];
   print join ' ', @{$t2[$i]}[@c2], "n" if $t2[$i];
}