First off a few general coding tips:

Always use the three parameter version of open, always check the result and always lexical file handles. Taking all that into account an open statement becomes:

open my $file1In, '<', $file1 or die "Failed to open $file1: $!";
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Always use strictures (use strict; use warnings;). if (($pr1{$key1} = $sil{$key._ref1}) ... may be a copy error, but it is wrong on two counts. Perl's numeric equality operator is == (assignment within an if expression is a really bad idea btw), and you actually want a string compare, not a numeric compare, so you should be using eq.

Something like the following may be what you want:

use strict;
use warnings;

my $file1 = <<END_FILE1;
Bbbb,3333,4444
Cdd,3444,3444
Zddd,345,456
END_FILE1

my $file2 = <<END_FILE2;
Bbbb_rf1,23,45
Cdd_rf2,24.45
Zddd,34,456.4
END_FILE2

my %concat;

open my $file1In, '<', \$file1 or die "Failed to open file1: $!";
while (<$file1In>) {
    chomp;
    my ($key, @tail) = split ',';
    
    next if ! @tail;
    $concat{$key} = \@tail;
}
close $file1In;

open my $file2In, '<', \$file2 or die "Failed to open file2: $!";
while (<$file2In>) {
    chomp;
    my ($key, @tail) = split ',';
    
    next if ! @tail or $key !~ s/_(\w*)$// or $1 ne 'rf1';
    push @{$concat{$key}}, @tail;
    print join (',', $key, @{$concat{$key}}), "\n";
}
close $file2In;
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Prints:

Bbbb,3333,4444,23,45
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